Control analysis and synthesis for general control-affine systemsThanks:

Cyprien Tamekue Thanks: Department of Electrical and Systems Engineering, Washington University in St. Louis, St. Louis, 63130, MO, USA. (cyprien@wustl.edu, shinung@wustl.edu). The first author thanks his former advisor, Yacine Chitour, for many relevant ideas and advice. ShiNung Ching^†^†footnotemark:

Abstract

We study controllability and constructive synthesis for control–affine systems. We introduce trajectory–dependent Gramian maps that extend the linear time–varying Gramian and yield explicit fixed–point synthesis maps. On feasible coercivity classes (uniform eigenvalue lower bounds), the Gramian map is Lipschitz, and under a comparison estimate criterion, synthesis iterates exhibit decay, and the Banach fixed-point theorem gives a unique fixed point that steers the system and satisfies an energy identity. When, in addition, an orthogonality condition holds, this fixed point coincides with the unique global minimum–energy control on the feasible set; if the coercivity bound holds uniformly for all bounded controls, the same conclusion holds on the full bounded–control space. We provide structural conditions on the input matrix that ensure the nonemptiness of the feasible class (and, in fully actuated regimes, equality with the full space) and sufficient conditions for underactuated systems via bounded–amplitude reference controls. Case studies on Hopfield network dynamics illustrate refined estimates that enlarge reachable targets. A trajectory–freezing and compactness step extends the synthesis to general nonlinear control-affine systems. The results yield verifiable controllability criteria with explicit, numerically implementable controllers.

keywords

Control-affine systems; nonlinear controllability; fixed-point methods; controllability Gramians; reachability; minimal-energy control.

Funding.

This work is partially supported by grant R21MH132240 from the US National Institutes of Health to SC.

^†^†runningheads: Control analysis and synthesis for general control-affine systems / C. Tamekue and S. Ching

MSC

93B05, 93C10, 93C15, 49K15, 49N35.

1 Introduction

We study control analysis and implementable synthesis for the control–affine class

\dot{x}(t)=A(t,x(t))\,N_{t}(x(t))+B(t,x(t))\,u(t),\qquad x(t_{0})=x^{0},\quad t\in[t_{0},T],

(1.1)

where at time $t$ , the system’s state is $x(t)=(x_{1}(t),\ldots,x_{d}(t))^{\top}\in{\mathbb{R}}^{d}$ , $x^{0}=(x_{1}^{0},\ldots,x_{d}^{0})^{\top}\in{\mathbb{R}}^{d}$ is the initial state, $u(t)=(u_{1}(t),\ldots,u_{k}(t))^{\top}\in{\mathbb{R}}^{k}$ is the control input, $N_{t}$ is a nonautonomous, vector field, $A(t,x)$ and $B(t,x)$ are time- and state-dependent matrices. Models of this form cover, among others, recurrent neural dynamics [15, 24] and standard engineering systems [6], and hence their study remains important.

Systems of the form (1.1) have been extensively studied in control theory, with a wide range of mathematical techniques developed over the past decades. Classical approaches include linearization-based analysis [25, 10], homotopy continuation [8, 17, 16, 29], and the return method [9, 10], which remains one of the most powerful constructive tools for global controllability. Geometric and algebraic methods based on Lie algebras provide structural controllability criteria [14, 20, 21, 27, 28, 10], while topological arguments offer complementary perspectives [12, 24]. Other analytic techniques include power series expansions [10] and fixed-point approaches [33, 34, 10, 20]. These contributions have led to a rich theory of local and global controllability; yet, implementable minimum-energy controls remain scarce beyond the linear time-invariant (LTI) and linear time-varying (LTV) settings [18], where the controllability Gramian yields closed-form minimum-energy controls.

Beyond the LTI/LTV setting, explicit Gramian–based formulas for minimum–energy controls are rarely available, and many nonlinear controllability results are qualitative or rely on continuation along a state path. Our stance is to recover “Gramian calculus” for system (1.1) by introducing trajectory–dependent controllability Gramian maps and casting synthesis as a fixed–point problem on an feasible coercivity class. This yields (i) a constructive control law via an implicit Gramian formula; (ii) a built–in energy certificate; and (iii) a convergence mechanism (via explicit Volterra rank-one comparison estimates for the iterates of the synthesis map), which is directly implementable. In contrast, the homotopy continuation method (HCM) [8, 17, 29] advances by integrating the path lifting equation and therefore requires surjectivity of the differential of the endpoint map along the entire path and typically offers no intrinsic energy certificate. Note, however, that a regularized continuation method has been proposed in [22] to deal with the case where this differential is not surjective. Still, the approach requires some conditions for the solvability: the drift dynamics must be well-posed over $[t_{0},T]$ , and the family of regularized path lifting equations must converge to a solution of the original problem. See also [16, 23]. Our framework trades this global rank condition for verifiable, local requirements: the coercivity of a trajectory–dependent Gramian, a self–mapping inequality on the feasible class, and an explicit iterate-contractivity condition. In practice, this matters when complete controllability is not known a priori: the method still furnishes steering controls (and their cost) whenever the feasible class is nonempty, and it delivers a quantitative subset of the reachable set. When HCM does apply, the two approaches are complementary—the nonlinear Gramian recovers the classical solution in the linear case and can serve as a preconditioner or initializer for continuation—while in regimes where the rank fluctuates, the Banach fixed–point route remains robust by construction.

Our main contributions are the following: (i) Baseline synthesis on $\dot{x}=N_{t}(x)+B(t,x)u$ : We work on a feasible coercivity class ${\mathcal{F}}(C)$ (uniform lower bound on the trajectory–dependent Gramian). On ${\mathcal{F}}(C)$ , the synthesis operator is well defined, and—on a self–mapped ball—a Volterra rank-one comparison estimate gives an iterate-contractivity criterion, which yields a unique fixed point with the Picard convergence and an energy certificate. (ii) From relative to global minimality: If an orthogonality condition holds at the fixed point, then this fixed point is the unique global $L^{2}$ –minimizer over the feasible set. (iii) Structural guarantees: An integral nondegeneracy of the input matrix $B$ implies uniform coercivity ( ${\mathcal{F}}(C)=L^{\infty}$ ). More generally, a reference control $u^{\flat}$ with invertible Gramian and amplitude small relative to a prescribed model-driven constant ensures the self–map property for explicit target radii; model structure (e.g., Hopfield networks) sharpens constants and enlarges admissible targets. (iv) Reachability: Self–mapping on ${\mathcal{F}}(C)$ together with the iterate-contractivity condition provides constructive steering with quantitative energy bounds; if ${\mathcal{F}}(C)=L^{\infty}$ one recovers complete controllability, otherwise one obtains a computable subset of the reachable set. (v) Extension to the general system (1.1): Freezing along an auxiliary trajectory reduces (1.1) to a family of baseline problems with uniform Lipschitz control of the Gramian map; Schaefer’s theorem then yields steering controls for admissible targets. For $N_{t}(x)=x$ , the construction recovers [10, Theorem. 3.40].

The remainder of the paper is structured as follows. In Notation 1 and Assumptions 2, we introduce the notation and assumptions used throughout. Section 3 develops the synthesis framework for the baseline system $\dot{x}=N_{t}(x)+B(t,x)u$ : we introduce the nonlinear Gramian representations, construct the synthesis maps, define the feasible coercivity class, and use an iterate-contractivity condition to obtain steering controls. Structurally sufficient conditions are then established, and refined estimates are illustrated on Hopfield-type networks. Section 4 extends the framework to the general control-affine system (1.1) by freezing along auxiliary trajectories and invoking a compactness argument to recover reachability for admissible targets. Section 5 summarizes the results and discusses perspectives for future work, and main proofs are collected in Section A.

Notation 1.

In the following, we denote by ${\mathbb{R}}^{d\times k}$ the set of $d\times k$ matrices with real coeﬃcients. ${\mathbb{R}}^{d}$ denotes the vector of $d$ real columns in dimension and ${\mathcal{L}}({\mathbb{R}}^{k},{\mathbb{R}}^{d})$ the space of linear maps from ${\mathbb{R}}^{k}$ to ${\mathbb{R}}^{d}$ that we identify, in the usual way, with ${\mathbb{R}}^{d\times k}$ . For all $(x,y)\in{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}$ , we denote by $|x|$ the Euclidean norm of $x$ and by $\langle x,y\rangle$ the scalar product of $x$ and $y$ . For a symmetric positive definite matrix $M\in{\mathbb{R}}^{d\times d}$ , the norm of $x\in{\mathbb{R}}^{d}$ with respect to $M$ is defined by $|x|_{M}^{2}=x^{\top}Mx$ . We denote the identity matrix of any size by ${\operatorname{Id}}$ , and for every $M\in{\mathbb{R}}^{d\times k}$ , $\|M\|=\sup\{|Mx|:x\in{\mathbb{R}}^{k},\;|x|=1\}$ denotes the spectral norm of $M$ . For a symmetric matrix $M\in{\mathbb{R}}^{d\times d}$ , $\lambda_{\min}(M)$ and $\lambda_{\max}(M)$ denote respectively its smaller and largest eigenvalues. Finally, we use $\mathscr{L}(X,Y)$ to denote the space of linear bounded operators from two normed vector spaces $X$ and $Y$ .

Assumption 2.

Throughout the following, unless otherwise stated, the nonautonomous vector field $N_{t}:[t_{0},T]\times{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ satisfies the following assumptions

1.

The map $t\mapsto N_{t}(x)$ belongs to $L^{\infty}(t_{0},T)$ for every fixed $x\in{\mathbb{R}}^{d}$ ,
2.

The map $x\mapsto N_{t}(x)$ is $C^{2}$ for every fixed $t\in[t_{0},T]$ .

Additionaly, for every $t\in[t_{0},T]$ , $N_{t}$ is globally $\Lambda_{1}$ -Lipschitz function for some $\Lambda_{1}>0$ , viz.

\|DN_{t}(w)\|\leq\Lambda_{1}\qquad\forall(t,w)\in[t_{0},T]\times{\mathbb{R}}^{d},

(1.2)

where $DN_{t}(w)\in{\mathbb{R}}^{d\times d}$ is the differential of $N_{t}$ at any $w\in{\mathbb{R}}^{d}$ . Finally, we also assume that its second derivative $D^{2}N_{t}$ satisfy for some $\Lambda_{2}>0$

\|D^{2}N_{t}(x)\|\leq\Lambda_{2}\qquad\forall(t,x)\in[t_{0},T]\times{\mathbb{R}}^{d}.

(1.3)

Remark 3.

For every fixed $t\in[t_{0},T]$ , the map $x\mapsto N_{t}(x)$ does not need to be bounded. Throughout the manuscript, we let $\Delta t_{0}:=T-t_{0}$ .

2 Prerequisites and flow representation

It follows from Assumption 2 that the nonautonomous flow $t\in[t_{0},T]\mapsto\Phi_{t_{0},t}$ associated with $N_{t}$ is globally Lipschitz and solves the following equation [4, Chapter 2]

\frac{d}{dt}\Phi_{t_{0},t}(x^{0})=N_{t}(\Phi_{t_{0},t}(x^{0})),\qquad\Phi_{t_{0},t_{0}}(x^{0})=x^{0}\in{\mathbb{R}}^{d}.

(2.1)

Moreover, $\{\Phi_{s,t}\mid(s,t)\in[t_{0},T]^{2}\}$ forms a two-parameter family of diffeomorphisms which satisfies the following algebraic identities

$\displaystyle\Phi_{t,t}$	$\displaystyle={\operatorname{Id}}\qquad\forall t\in[t_{0},T],$	(2.2)
$\displaystyle\Phi_{t_{2},t_{3}}\circ\Phi_{t_{1},t_{2}}$	$\displaystyle=\Phi_{t_{1},t_{3}}\qquad\forall(t_{1},t_{2},t_{3})\in[t_{0},T]^{3},$	(2.3)
$\displaystyle(\Phi_{t_{1},t_{2}})^{-1}$	$\displaystyle=\Phi_{t_{2},t_{1}}\qquad\forall(t_{1},t_{2})\in[t_{0},T]^{2}.$	(2.4)

Furthermore, for fixed $(s,t)\in[t_{0},T]^{2}$ , the maps $\Phi_{s,t}$ and $\Phi_{t,s}$ are differentiable at every $x^{0}$ and $\Phi_{s,t}(x^{0})$ , respectively. Denoting these differential by $D\Phi_{s,t}(x^{0})$ , and $D\Phi_{t,s}(\Phi_{s,t}(x^{0}))$ , they solve

\begin{cases}\displaystyle\frac{d}{dt}D\Phi_{s,t}(x^{0})&=DN_{t}(\Phi_{s,t}(x^{0}))\,D\Phi_{s,t}(x^{0}),\qquad D\Phi_{s,s}(x^{0})={\operatorname{Id}},\\ \cr\displaystyle\frac{d}{dt}D\Phi_{t,s}(\Phi_{s,t}(x^{0}))&=-D\Phi_{t,s}(\Phi_{s,t}(x^{0}))\,DN_{t}(\Phi_{s,t}(x^{0})),\qquad D\Phi_{s,s}(\Phi_{s,t}(x^{0}))={\operatorname{Id}}.\end{cases}

(2.5)

In particular, the following holds for every $x^{0}\in{\mathbb{R}}^{d}$ ,

[D\Phi_{s,t}(x^{0})]^{-1}=D\Phi_{t,s}(\Phi_{s,t}(x^{0}))\qquad\forall(s,t)\in[t_{0},T]^{2}.

(2.6)

We collect useful estimates related to $\Phi_{s,t}$ and $\Phi_{t,s}$ in the following lemma. The proof uses the standard Gronwall lemma.

Lemma 4.

Let $\beta\in C^{0}([t_{0},T];{\mathbb{R}}^{d})$ and set $q:=\Lambda_{2}/\Lambda_{1}$ . It holds for every $(s,t)\in[t_{0},T]^{2}$ , $s\leq t$ ,

\begin{split}\|D\Phi_{s,t}(\beta(s))\|&\leq e^{\Lambda_{1}(t-s)},\qquad\|D\Phi_{t,s}(\beta(s))\|\leq e^{\Lambda_{1}(t-s)},\\ \|D\Phi_{s,t}(\beta(t))\|&\leq e^{\Lambda_{1}(t-s)},\qquad\|D\Phi_{t,s}(\beta(t))\|\leq e^{\Lambda_{1}(t-s)},\end{split}

(2.7)

\begin{split}\|D^{2}\Phi_{s,t}(\beta(s))\|&\leq q(e^{2\Lambda_{1}(t-s)}-e^{\Lambda_{1}(t-s)}),\qquad\|D^{2}\Phi_{t,s}(\beta(s))\|\leq q(e^{2\Lambda_{1}(t-s)}-e^{\Lambda_{1}(t-s)}),\\ \|D^{2}\Phi_{s,t}(\beta(t))\|&\leq q(e^{2\Lambda_{1}(t-s)}-e^{\Lambda_{1}(t-s)}),\qquad\|D^{2}\Phi_{t,s}(\beta(t))\|\leq q(e^{2\Lambda_{1}(t-s)}-e^{\Lambda_{1}(t-s)}),\end{split}

(2.8)

where $D^{2}\Phi_{s,t}$ is the second derivative (third-order tensor) of $\Phi_{s,t}$ .

3 Control–affine dynamics with state–dependent and time-varying input matrix

In this section, we consider the following control–affine system with state–dependent and time-varying input matrix

\dot{x}(t)=N_{t}(x(t))+B(t,x(t))u(t),\quad x(t_{0})=x^{0},\quad t\in[t_{0},T],

(

\Sigma

)

which is a specific case of (1.1) where $A(t,x)={\operatorname{Id}}$ . We recall that $(t,x)\mapsto N_{t}(x)$ satisfies Assumption 2.

Assumption 5.

The input matrix $B:[t_{0},T]\times{\mathbb{R}}^{d}\to{\mathbb{R}}^{d\times k}$ is an element of $L^{\infty}((t_{0},T)\times{\mathbb{R}}^{d};{\mathbb{R}}^{d\times k})$ . Furthermore, the map $x\mapsto B(t,x)$ is $C^{1}$ and globally Lipschitz for every $t\in[t_{0},T]$ , i.e., there exists $L_{B}\geq 0$ such that

\|D_{x}B(t,w)\|\leq L_{B}\qquad\forall(t,w)\in[t_{0},T]\times{\mathbb{R}}^{d}.

(3.1)

Note that if $B(t,\cdot)=B(t)$ , then $L_{B}=0$ .

3.1 Representation of solutions

The existence and uniqueness of global solutions to ( Σ ) follow as a straightforward application of the Cauchy-Lipschitz theorem, given Assumptions 2 and 5, see, for instance, [4, Theorem 3.2.1]. However, the main objective of this paper is to synthesize a control function that solves the associated two-point boundary value problem. To this end, leveraging the regularity properties of the drift $N_{t}$ , we consider solution representations of ( Σ ) that facilitate control design. These solution representations fall within the general framework of chronological calculus introduced in [1] and leveraged in [19, 21] for applications in geometric control theory. See also [2, Chapter 6] for a comprehensive presentation of this theory. In [30], solution representations of nonlinear control systems like ( Σ ) were established when $N_{t}=N$ , $B(t,x)=B$ , and the control was constant; see also the proof of [31, Theorem 4.4]. In contrast, the present work extends these representations to the framework of ( Σ ). For completeness, we replace in ( Σ ), $B(t,x)\,u(t)$ with $b(t,x)\in L^{\infty}((t_{0},T)\times{\mathbb{R}}^{d};{\mathbb{R}}^{d})$ where $x\mapsto\,b(t,x)$ is locally Lipschitz. The proof of the following theorem is given in Section A.1.

Theorem 6.

For all $x^{0}\in{\mathbb{R}}^{d}$ , and $b\in L^{\infty}((t_{0},T)\times{\mathbb{R}}^{d};{\mathbb{R}}^{d})$ such that $x\mapsto\,b(t,x)$ is locally Lipschitz, the system $\dot{x}(t)=N_{t}(x(t))+b(t,x(t))$ , $x(t_{0})=x^{0}$ admits a unique absolutely continuous solution $x\in C^{0}([t_{0},T];{\mathbb{R}}^{d})$ that can be written for $i\in\{1,2\}$ ,

x(t)=\Phi_{\tau_{i},t}\!\left(\Phi_{t_{0},\tau_{i}}(x^{0})+\int_{t_{0}}^{t}D\Phi_{s,\tau_{i}}\big(x(s)\big)\,b(s,x(s))\,ds\right),\qquad t\in[t_{0},T],\quad\tau_{1}=t_{0},\;\tau_{2}=T.

(3.2)

The representation corresponding to $\tau_{1}=t_{0}$ is termed forward representation, while that corresponding to $\tau_{2}=T$ is termed backward representation, consistent with their temporal directionality.

Observe that if $N_{t}\equiv A(t)$ , where $A:[t_{0},T]\to\mathbb{R}^{d\times d}$ is a matrix-valued function in $L^{\infty}((t_{0},T);\mathbb{R}^{d\times d})$ , then (3.2) reduces to the classical representation of solutions for linear time-varying (LTV) systems.

Corollary 7.

If $N_{t}\equiv A(t)\in L^{\infty}((t_{0},T);{\mathbb{R}}^{d\times d})$ and $b(t,\cdot)\equiv b(t)\in L^{\infty}((t_{0},T);{\mathbb{R}}^{d})$ , then (3.2) recasts as

x(t)=R(t,t_{0})\,x^{0}+\int_{t_{0}}^{t}R(t,s)\,b(s)\,ds\qquad\forall t\in[t_{0},T],

(3.3)

where $R(t,s)\in C^{0}([t_{0},T]^{2};{\mathbb{R}}^{d\times d})$ is the state-transition matrix of $\dot{y}(t)=A(t)\,y(t)$ satisfying $R(s,s)={\operatorname{Id}}$ .

Proof.

First, by (2.1)-(2.4) and the definition of $R(t,s)$ , one has $\Phi_{s,t}(x)=R(t,s)\,x$ for all $x\in{\mathbb{R}}^{d}$ . It follows that $D\Phi_{s,t}(x(t))=R(t,s)$ for all $(t,s)\in[t_{0},T]^{2}$ . Therefore, from (3.2), one deduces

x(t)=\Phi_{\tau_{i},t}\left(\Phi_{t_{0},\tau_{i}}(x^{0})+\int_{t_{0}}^{t}\!\!D\Phi_{s,\tau_{i}}(x(s))\,b(s)\,ds\right)=R(t,t_{0})\,x^{0}+R(t,\tau_{i})\int_{t_{0}}^{t}\!\!R(\tau_{i},s)\,b(s)\,ds

(3.4)

which yields (3.3).

We now establish results that relate the state-transition matrix of the linearized form of control system ( Σ ) to derivatives of the flow of the vector field $N_{t}$ along the trajectory. It will be essential for our subsequent results. The proof is presented in Section A.2.

Lemma 8.

Let $(x^{0},u)\in{\mathbb{R}}^{d}\times L^{\infty}((t_{0},T);{\mathbb{R}}^{k})$ , and let $x_{u}\in C^{0}([t_{0},T];{\mathbb{R}}^{d})$ be the solution of ( Σ ). Then, the state-transition matrix $R_{u}(t,s)\in C^{0}([t_{0},T]^{2};{\mathbb{R}}^{d\times d})$ of the linearized equation

\dot{y}(t)=\big[DN_{t}(x_{u}(t))+D_{x}B(t,x_{u}(t))u(t)\big]y(t),\qquad t\in[t_{0},T]

(3.5)

satisfying $R_{u}(s,s)={\operatorname{Id}}$ , is given by $R_{u}(t,s)=P_{\tau_{i},u}(t,s)$ , $i\in\{1,2\}$ where $P_{\tau_{i},u}(t,s)$ can be factorized as

P_{\tau_{i},u}(t,s):=\big[D\Phi_{t,\tau_{i}}(x_{u}(t))\big]^{-1}\,M_{\tau_{i},u}(t,s)\,D\Phi_{s,\tau_{i}}(x_{u}(s)),\qquad(t,s)\in[t_{0},T]^{2},\quad\tau_{1}=t_{0},\,\tau_{2}=T.

(3.6)

Here, $M_{\tau_{i},u}(t,s)\in C^{0}([t_{0},T]^{2};{\mathbb{R}}^{d\times d})$ satisfying $M_{\tau_{i},u}(s,s)={\operatorname{Id}}$ is the state-transition matrix of

\dot{y}(t)={\mathcal{A}}_{\tau_{i},u}(t)\,y(t),\qquad t\in[t_{0},T]

(3.7)

where ${\mathcal{A}}_{\tau_{i},u}(t):=\big[D^{2}\Phi_{t,\tau_{i}}(x_{u}(t))B(t,x_{u}(t))u(t)+D\Phi_{t,\tau_{i}}\big(x_{u}(t)\big)D_{x}B\big(t,x_{u}(t)\big)u(t)\big]\big[D\Phi_{t,\tau_{i}}(x_{u}(t))\big]^{-1}\in{\mathbb{R}}^{d\times d}$ .

Remark 9.

When $B(t,x)\equiv B(t)\in L^{\infty}((t_{0},T);{\mathbb{R}}^{k})$ is state-independent, it is clear from the factorization (3.6) that the state-transition matrix of $\dot{y}(t)=DN_{t}(x_{u}(t))y(t)$ is different from $D\Phi_{s,t}(x_{u}(s))$ in general, and that there is equality if the control $u=0$ or if $N_{t}$ is linear. In the former case, $M_{\tau_{i},0}(t,s)={\operatorname{Id}}$ , and we use

S_{t_{0},x^{0}}(t,s):=P_{t_{0},0}(t,s)=D\Phi_{t_{0},t}(x^{0})\,D\Phi_{s,t_{0}}(\Phi_{t_{0},s}(x^{0}))=D\Phi_{s,t}(\Phi_{t_{0},s}(x^{0}))\qquad\forall(s,t)\in[t_{0},T]^{2}

(3.8)

as the representation of the state-transition matrix of the linearized equation $\dot{y}(t)=DN_{t}(\Phi_{t_{0},t}(x^{0}))\,y(t)$ .

3.2 Controllability results

Given $(x^{0},x^{1})\in{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}$ , our goal in this section is to identify sufficient conditions under which system ( Σ ) can be steered from the initial state $x^{0}$ to the target $x^{1}$ over the horizon $[t_{0},T]$ . Whenever these conditions are met, we provide a synthesis of two feasible controls realizing this transfer.

3.2.1 Synthesis, analysis, and optimal control results

As announced, we present our control synthesis, analysis, and optimal control results in this section. We start with recalling the following

Definition 10.

Given $x^{0}\in{\mathbb{R}}^{d}$ , we say that $x^{1}\in{\mathbb{R}}^{d}$ is reachable on $[t_{0},T]$ from $x^{0}$ if there exists a control $u\in L^{\infty}((t_{0},T);{\mathbb{R}}^{k})$ such that the solution $x_{u}(\cdot)$ of ( Σ ) satisfies $x_{u}(T)=x^{1}$ . The reachable set is denoted by

{\mathcal{R}}_{\Sigma}(T,x^{0}):=\Big\{\,x_{u}(T)\;:\;x_{u}(\cdot)\ \text{solves \eqref{eq:state-dependent-input} with }x(t_{0})=x^{0}\,\Big\}.

(3.9)

If ${\mathcal{R}}_{\Sigma}(T,x^{0})={\mathbb{R}}^{d}$ for every $x^{0}\in{\mathbb{R}}^{d}$ , one says that ( Σ ) is completely controllable on $[t_{0},T]$ .

For a fixed $x^{0}\in{\mathbb{R}}^{d}$ and control $u\in L^{\infty}((t_{0},T);{\mathbb{R}}^{k})$ , let $x_{u}(\cdot):=x_{u,x^{0}}(\cdot)$ be the corresponding trajectory of ( Σ ). We associate with $u$ and $x^{0}$ the matrices ( $\tau_{1}=t_{0}$ and $\tau_{2}=T$ )

{\mathcal{N}}_{\tau_{i}}(u):={\mathcal{N}}_{\tau_{i}}(u,x^{0})=\int_{t_{0}}^{T}D\Phi_{t,\tau_{i}}(x_{u}(t))\,B(t,x_{u}(t))\,B(t,x_{u}(t))^{\top}\,D\Phi_{t,\tau_{i}}(x_{u}(t))^{\top}\,dt,\quad\,i\in\{1,2\}.

(3.10)

Remark 11.

(1) If $N_{t}\equiv A(t)\in L^{\infty}((t_{0},T);{\mathbb{R}}^{d\times d})$ and $B(t,\cdot)\equiv B(t)\in L^{\infty}((t_{0},T);{\mathbb{R}}^{d\times k})$ , then ${\mathcal{N}}_{\tau_{1}}(u)$ is congruent to the controllability Gramian $W_{c}$ of the LTV system $\dot{x}(t)=A(t)\,x(t)+B(t)\,u(t)$ , and ${\mathcal{N}}_{\tau_{2}}(u)=W_{c}$ .

(2) For any $y\in{\mathbb{R}}^{d}$ ,

y^{\top}{\mathcal{N}}_{\tau_{i}}(u)\,y=\int_{t_{0}}^{T}|y^{\top}D\Phi_{t,\tau_{i}}(x_{u}(t))\,B(t,x_{u}(t))|^{2}\,dt,

(3.11)

hence ${\mathcal{N}}_{\tau_{i}}(u)$ is symmetric and positive semi-definite (PSD).

(3) The Gramian map¹¹ 1 We use ${\mathcal{S}}_{d}^{+}({\mathbb{R}})$ to denote the set of symmetric positive semi-definite matrices of order $d$ with real coefficients. ${\mathcal{N}}_{\tau_{i}}:L^{\infty}((t_{0},T);{\mathbb{R}}^{k})\to{\mathcal{S}}_{d}^{+}({\mathbb{R}})$ defined by (3.10) is in general different from the notion of Gramian used in the homotopy continuation method [8, 16, 29] where the Gramian is defined as

M(u):=\int_{t_{0}}^{T}\,R_{u}(T,t)\,B(t,x_{u}(t))\,B(t,x_{u}(t))^{\top}\,R_{u}(T,t)^{\top}\,dt,\qquad\,u\in L^{\infty}((t_{0},T);{\mathbb{R}}^{k})

(3.12)

where $R_{u}(t,s)$ is defined in Lemma 8.

The proof of the following proposition uses standard arguments of minimization in Hilbert spaces.

Proposition 12.

Let $x^{0}\in{\mathbb{R}}^{d}$ , $u\in L^{\infty}((t_{0},T);{\mathbb{R}}^{k})$ , and $x_{u}\in C^{0}([t_{0},T];{\mathbb{R}}^{d})$ be the solution to ( Σ ). Fix $i\in\{1,2\}$ and define the bounded linear operator $L_{u,\tau_{i}}:L^{2}((t_{0},T);{\mathbb{R}}^{k})\to{\mathbb{R}}^{d}$ by

L_{u,\tau_{i}}v=\int_{t_{0}}^{T}D\Phi_{t,\tau_{i}}(x_{u}(t))\,B(t,x_{u}(t))\,v(t)\,dt,\qquad\tau_{1}:=t_{0},\,\tau_{2}:=T.

(3.13)

For $y_{i}\in\operatorname{Ran}(L_{u,\tau_{i}})$ , the problem

\min\{\|v\|_{2}:L_{u,\tau_{i}}v=y_{i}\}

(3.14)

has the solution $v_{i}(u):=v_{i}(u,x^{0},y_{i})$ given by

v_{i}(u)(t)=\left[L_{u,\tau_{i}}^{\ast}\,{\mathcal{N}}_{\tau_{i}}(u)^{\dagger}\,y_{i}\right](t)=B(t,x_{u}(t))^{\top}D\Phi_{t,\tau_{i}}(x_{u}(t))^{\top}\,{\mathcal{N}}_{\tau_{i}}(u)^{\dagger}\,y_{i},\qquad t\in[t_{0},T].

(3.15)

Here $L_{u,\tau_{i}}^{\ast}$ is the adjoint of $L_{u,\tau_{i}}$ , and ${\mathcal{N}}_{\tau_{i}}(u)^{\dagger}$ denotes the Moore–Penrose pseudoinverse of ${\mathcal{N}}_{\tau_{i}}(u)$ .

Proposition 12 ensures that $v_{i}(u)$ is $L^{2}$ –minimal within the affine subspace defined by $L_{u,\tau_{i}}$ . In the nonlinear setting, these subspaces depend on $u$ , so minimality is relative rather than global.

Using (2.7) and the singular value decomposition of symmetric PSD matrices, one finds that

\|v_{i}(u)\|_{\infty}\leq\frac{\|B\|_{\infty}e^{\Lambda_{1}\Delta t_{0}}|y_{i}|}{\lambda_{\min}^{\ast}({\mathcal{N}}_{\tau_{i}}(u))}\qquad\forall u\in L^{\infty}((t_{0},T);{\mathbb{R}}^{k}),\ i\in\{1,2\},

(3.16)

where $\lambda_{\min}^{\ast}({\mathcal{N}}_{\tau_{i}}(u))>0$ is the smallest nonzero eigenvalue of ${\mathcal{N}}_{\tau_{i}}(u)$ .

Feasible coercivity class

Let $\lambda_{\min}\big({\mathcal{N}}_{\tau_{i}}(u)\big)$ denote the smallest eigenvalue of ${\mathcal{N}}_{\tau_{i}}(u)$ , and let $C_{i}>0$ be a constant depending only on system data (e.g. $\|B\|_{\infty}$ , $\Lambda_{1}$ , $\Lambda_{2}$ , $\Delta t_{0}$ , and possibly $x^{0}$ ). Guided by (3.16), we encode the positivity of ${\mathcal{N}}_{\tau_{i}}(u)$ through the feasible coercivity class

{\mathcal{F}}(C_{i}):=\Big\{u\in L^{\infty}((t_{0},T);{\mathbb{R}}^{k}):\ \lambda_{\min}\big({\mathcal{N}}_{\tau_{i}}(u)\big)\geq C_{i}^{-1}\Big\},\quad i\in\{1,\,2\}.

(3.17)

The following observations on ${\mathcal{F}}(C_{i})$ are immediate.

Remark 13.

1.

Possibility of emptiness. For a given $C_{i}>0$ , the feasible coercivity class ${\mathcal{F}}(C_{i})$ may be empty. Non-emptiness requires the existence of at least one control $u$ with $\lambda_{\min}({\mathcal{N}}_{\tau_{i}}(u))\geq C_{i}^{-1}$ .
2.

Monotonicity in $C_{i}$ . If $C_{i}\leq C_{i}^{\prime}$ , then ${\mathcal{F}}(C_{i})\subseteq{\mathcal{F}}(C_{i}^{\prime})$ .
3.

Topological property. By Assumptions 2, the map $u\mapsto{\mathcal{N}}_{\tau_{i}}(u)$ is continuous, hence $u\mapsto\lambda_{\min}({\mathcal{N}}_{\tau_{i}}(u))$ is continuous. Therefore ${\mathcal{F}}(C_{i})$ is closed in $L^{\infty}((t_{0},T);{\mathbb{R}}^{k})$ . We clarify this point later in Lemma 25 and Corollary 27.
4.

Non-emptiness criterion. If there exists a control reference $u^{\flat}\in L^{\infty}((t_{0},T);{\mathbb{R}}^{k})$ with $\lambda_{\min}({\mathcal{N}}_{\tau_{i}}(u^{\flat}))\geq C_{i}^{-1}$ , then ${\mathcal{F}}(C_{i})\neq\emptyset$ . By the continuity of $u\mapsto\lambda_{\min}({\mathcal{N}}_{\tau_{i}}(u))$ , a neighborhood of $u^{\flat}$ also lies in ${\mathcal{F}}(C_{i})$ . A sufficient condition ensuring ${\mathcal{F}}(C_{i})=L^{\infty}((t_{0},T);{\mathbb{R}}^{k})$ for every $x^{0}\in{\mathbb{R}}^{d}$ is provided in Proposition 23.

For any $u\in{\mathcal{F}}(C_{i})$ , estimate (3.16) yields the uniform bound

\|v_{i}(u)\|_{\infty}\;\leq\zeta(y_{i})=C_{i}\,\|B\|_{\infty}\,e^{\Lambda_{1}\Delta t_{0}}\,|y_{i}|,\qquad i\in\{1,2\}.

(3.18)

Let $v_{i}(u,x^{0},y_{i})$ is defined as in Proposition 12, we define the synthesis maps

{\mathcal{S}}_{i}:L^{\infty}((t_{0},T);{\mathbb{R}}^{k})\to L^{\infty}((t_{0},T);{\mathbb{R}}^{k}),\;u\mapsto v_{i}(u,x^{0},y_{i}),\qquad y_{i}\in{\mathbb{R}}^{d},\quad i\in\{1,\,2\}.

(3.19)

Motivated by (3.18), we introduce the feasibility coercivity ball

{\mathcal{F}}(y_{i}):={\mathcal{F}}(C_{i})\cap{\mathcal{B}}_{i}\quad\text{where}\quad{\mathcal{B}}_{i}:=\{u\in L^{\infty}((t_{0},T);{\mathbb{R}}^{k}):\|u\|_{\infty}\leq\,\zeta_{i}:=\zeta(y_{i})\}.

(3.20)

Under the non-emptiness assumption on ${\mathcal{F}}(C_{i})$ , and if ${\mathcal{S}}_{i}$ acts as a self-map on ${\mathcal{F}}(C_{i})$ , namely

{\mathcal{S}}_{i}({\mathcal{F}}(C_{i}))\subset{\mathcal{F}}(C_{i}),

(3.21)

we can prove the existence and uniqueness of fixed points for ${\mathcal{S}}_{i}$ in ${\mathcal{F}}(y_{i})$ , whenever an associated Volterra rank-one comparison estimate is contractive.

The proof of the following result is given in Section A.3.

Theorem 14.

Fix $i\in\{1,2\}$ . Assume that ${\mathcal{F}}(C_{i})\neq\emptyset$ and that (3.21) is satisfied. Then there exists a positive sequence $(\varrho_{m})_{m\geq 1}$ with $\varrho_{m}:=K_{i}^{m}\gamma_{m}(p_{i})$ such that

\|{\mathcal{S}}_{i}^{\,m}(u)-{\mathcal{S}}_{i}^{\,m}(v)\|_{\infty}\leq\varrho_{m}\|u-v\|_{\infty},\qquad\forall\,u,v\in{\mathcal{F}}(y_{i}),\ \forall\,m\geq 1,

(3.22)

where $K_{i}>0$ depends only on system data and $C_{i}$ , and for some $p_{i}\in(0,1)$ and $\gamma_{m}(p_{i})\to 0$ as $m\to\infty$ . Furthermore, if there exists $m_{i}\geq 1$ such that

\varrho_{m_{i}}<1,

(3.23)

then ${\mathcal{S}}_{i}$ admits a unique fixed point $u_{i}\in{\mathcal{F}}(y_{i})$ , and the Picard iteration $u^{(n+1)}={\mathcal{S}}_{i}(u^{(n)})$ converges to $u_{i}$ for any $u^{(0)}\in{\mathcal{F}}(y_{i})$ .

Remark 15.

If ${\mathcal{F}}(C_{i})\neq\emptyset$ and (3.21) is satisfied, then ${\mathcal{F}}(y_{i})\neq\emptyset$ . In fact, under these assumptions, ${\mathcal{S}}_{i}(u)\in{\mathcal{F}}(y_{i})$ for all $u\in{\mathcal{F}}(C_{i})$ . Furthermore, if $B(t,x)\equiv B(t)$ is not state-dependent, the decay estimate (3.22) holds for all $(u,v)\in{\mathcal{F}}(C_{i})^{2}$ since the Gramian map ${\mathcal{N}}_{\tau_{i}}$ is Lipschitz continuous on the whole $L^{\infty}((t_{0},T);{\mathbb{R}}^{k})$ .

Theorem 14 guarantees the existence and uniqueness of a fixed point of the synthesis map ${\mathcal{S}}_{i}$ on ${\mathcal{F}}(y_{i})$ . To connect this fixed point with $L^{2}$ –norm minimality, we invoke a Lagrange multiplier. We will need preparatory results stated in Lemma 16 below. To this end, we introduce for any fixed $x^{0}\in{\mathbb{R}}^{d}$ the endpoint map

{\mathcal{E}}:={\mathcal{E}}_{x^{0},T}:u\in\,L^{\infty}((t_{0},T);{\mathbb{R}}^{k})\longmapsto{\mathbb{R}}^{d},\quad{\mathcal{E}}(u)=x_{u}(T)

(3.24)

and for $i\in\{1,2\}$ and any fixed $y_{i}\in{\mathbb{R}}^{d}$ , the feasible map

G_{\tau_{i}}:L^{\infty}((t_{0},T);{\mathbb{R}}^{k})\to{\mathbb{R}}^{d},\quad G_{\tau_{i}}(u)=L_{u,\tau_{i}}u-y_{i}

(3.25)

where $x_{u}(\cdot)$ is the solution of ( Σ ) and $L_{u,\tau_{i}}$ is defined in (3.13). The proof of the following is given in Section A.4.

Lemma 16.

One has ${\mathcal{E}},\,G_{\tau_{i}}\in C^{1}({\mathcal{Y}};{\mathbb{R}}^{d})$ . Furthermore, let $D{\mathcal{E}}(u),\,DG_{\tau_{i}}(u):L^{2}((t_{0},T);{\mathbb{R}}^{k})\to{\mathbb{R}}^{d}$ are, respectively the Fréchet derivative of ${\mathcal{E}}$ and $G_{\tau_{i}}$ at any fixed $u\in{\mathcal{Y}}$ . Then, one has the following identity

DG_{\tau_{i}}(u)=D\Phi_{T,\tau_{i}}\!\big(x_{u}(T)\big)\,D{\mathcal{E}}(u).

(3.26)

Moreover, let ${\mathcal{N}}_{\tau_{i}}(u)$ and $M(u):=D{\mathcal{E}}(u)\,D{\mathcal{E}}(u)^{\ast}$ are respectively defined in (3.10) and (3.12).

1.

If ${\mathcal{N}}_{\tau_{i}}(u)$ is invertible and ${\operatorname{Id}}+K_{u,\tau_{i}}L_{u,\tau_{i}}^{\ast}{\mathcal{N}}_{\tau_{i}}(u)^{-1}\in{\mathbb{R}}^{d\times d}$ is invertible, then $DG_{\tau_{i}}(u)L_{u,\tau_{i}}^{\ast}\in{\mathbb{R}}^{d\times d}$ is invertible and $DG_{\tau_{i}}(u)$ is right-invertible;
2.

If $M(u)$ is invertible and ${\operatorname{Id}}-K_{u,\tau_{i}}DG_{\tau_{i}}(u)^{\ast}M(u)^{-1}\in{\mathbb{R}}^{d\times d}$ is invertible, then $L_{u,\tau_{i}}DG_{\tau_{i}}(u)^{\ast}\in{\mathbb{R}}^{d\times d}$ is invertible and $L_{u,\tau_{i}}$ is right-invertible.

Here, we let $K_{u,\tau_{i}}:=DG_{\tau_{i}}(u)-L_{u,\tau_{i}}$ .

The following results connect the fixed-point of Theorem 14 with $L^{2}$ -nom minimality in the feasible set; its proof is presented in Section A.5.

Theorem 17.

Fix $i\in\{1,2\}$ , assume ${\mathcal{F}}(C_{i})\neq\emptyset$ , and ${\operatorname{Id}}+K_{u,\tau_{i}}L_{u,\tau_{i}}^{\ast}{\mathcal{N}}_{\tau_{i}}(u)^{-1}\in{\mathbb{R}}^{d\times d}$ is invertible for all $u\in{\mathcal{F}}(C_{i})$ . Let $y_{i}\in{\mathbb{R}}^{d}$ and the feasible set

\mathfrak{F}_{i}:=\{u\in{\mathcal{F}}(y_{i}):\ G_{\tau_{i}}(u)=0\}.

Then, the following are equivalent for any local minimizer $\bar{u}$ of $\tfrac{1}{2}\|u\|_{L^{2}}^{2}$ over $\mathfrak{F}_{i}$ :

1. $\bar{u}$ is a fixed point of ${\mathcal{S}}_{i}$ on ${\mathcal{F}}(y_{i})$ , i.e., $\bar{u}=L_{\bar{u},\tau_{i}}^{\ast}\,{\mathcal{N}}_{\tau_{i}}(\bar{u})^{-1}\,y_{i}$ ;

2. The following orthogonality condition holds

\langle[L_{\bar{u},\tau_{i}}DG_{\tau_{i}}(\bar{u})^{\ast}]^{-1}y_{i},DG_{\tau_{i}}(\bar{u})\,h\rangle_{{\mathbb{R}}^{d}}=0\qquad\forall\,h\in\ker L_{\bar{u},\tau_{i}}.

(3.27)

The following shows the existence of a local minimizer of $\tfrac{1}{2}\|u\|_{L^{2}}^{2}$ over $\mathfrak{F}_{i}$ . The proof is given in Section A.6.

Proposition 18.

Fix $i\in\{1,2\}$ and assume ${\mathcal{F}}(C_{i})\neq\emptyset$ . Let $y_{i}\in{\mathbb{R}}^{d}$ , then the feasible set

\mathfrak{F}_{i}:=\{u\in{\mathcal{F}}(y_{i}):\ G_{\tau_{i}}(u)=0\}

(3.28)

is weakly sequential closed in $L^{2}((t_{0},T);{\mathbb{R}}^{k})$ . Consequently, $\tfrac{1}{2}\|u\|_{L^{2}}^{2}$ attains a minimum on $\mathfrak{F}_{i}$ .

From Theorems 14, and 17, and Proposition 18, we deduce the synthesis and optimal control results:

Theorem 19.

Let $(x^{0},\,x^{1})\in{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}$ , $i\in\{1,2\}$ , and set $y_{i}:=\Phi_{T,\tau_{i}}(x^{1})-\Phi_{t_{0},\tau_{i}}(x^{0})$ . Assume ${\mathcal{F}}(C_{i})\neq\emptyset$ , (3.21) and (3.23) are satisfied, and let $u_{i}\in{\mathcal{F}}(y_{i})$ be the fixed point of ${\mathcal{S}}_{i}$ . Then $u_{i}$ steers ( Σ ) from $x^{0}$ to $x^{1}$ on $[t_{0},T]$ and

u_{i}(t)=L_{u_{i},\tau_{i}}^{\ast}\,{\mathcal{N}}_{\tau_{i}}(u_{i})^{-1}y_{i}=B(t,x_{u_{i}}(t))^{\top}D\Phi_{t,\tau_{i}}(x_{u_{i}}(t))^{\top}\,{\mathcal{N}}_{\tau_{i}}(u_{i})^{-1}y_{i},\quad t\in[t_{0},T],\quad\tau_{1}=t_{0},\,\tau_{2}=T,

(3.29)

with $\|u_{i}\|_{L^{2}}^{2}=y_{i}^{\top}\,{\mathcal{N}}_{\tau_{i}}(u_{i})^{-1}y_{i}$ .

If the hypotheses of Theorem 17 hold for every local minimizer over $\mathfrak{F}_{i}:=\{u\in{\mathcal{F}}(y_{i}):L_{u,\tau_{i}}u=y_{i}\}$ , and if (3.27) holds whenever needed, then $\tfrac{1}{2}\|u\|_{L^{2}}^{2}$ attains its minimum over $\mathfrak{F}_{i}$ , and the unique minimizer is $u_{i}$ . Equivalently,

\|u_{i}\|_{L^{2}}^{2}\leq\|w\|_{L^{2}}^{2}\quad\forall\,w\in\mathfrak{F}_{i},\qquad\text{with equality iff }w=u_{i}.

Proof.

Since $u_{i}={\mathcal{S}}_{i}(u_{i})=L_{u_{i},\tau_{i}}^{\ast}\,{\mathcal{N}}_{\tau_{i}}(u_{i})^{-1}y_{i}$ , we obtain (3.29), feasibility $L_{u_{i},\tau_{i}}u_{i}=y_{i}$ , and $\|u_{i}\|_{L^{2}}^{2}=y_{i}^{\top}{\mathcal{N}}_{\tau_{i}}(u_{i})^{-1}y_{i}$ . By Proposition 18, $\tfrac{1}{2}\|u\|_{L^{2}}^{2}$ attains a minimum on $\mathfrak{F}_{i}$ . Let $\bar{u}$ be a minimizer. Under the stated hypotheses, Theorem 17 implies that $\bar{u}$ is a fixed point of ${\mathcal{S}}_{i}$ . By uniqueness, $\bar{u}=u_{i}$ .

Remark 20 (A posteriori check of the minimizer).

When ${\mathcal{S}}_{i}$ admits a unique fixed point $u_{i}\in{\mathcal{F}}(y_{i})$ , it suffices to verify whether the orthogonality condition (3.27) holds at $u_{i}$ , viz.

\langle[L_{u_{i},\tau_{i}}DG_{\tau_{i}}(u_{i})^{\ast}]^{-1}y_{i},DG_{\tau_{i}}(u_{i})\,h\rangle_{{\mathbb{R}}^{d}}=0\qquad\forall\,h\in\ker L_{u_{i},\tau_{i}}.

(3.30)

In this case, Theorem 17 implies that $u_{i}$ is a local minimizer of $\frac{1}{2}\|u\|_{L^{2}}^{2}$ over $\mathfrak{F}_{i}$ , and that if the global minimizer $\bar{u}\in\mathfrak{F}_{i}$ satisfies (3.27), then it coincides with $u_{i}$ ; hence $u_{i}$ is the global minimum–energy control on $\mathfrak{F}_{i}$ .

Remark 21.

If $B(t,x)\equiv B(t)$ and there exists $C_{i}>0$ such that ${\mathcal{F}}(C_{i})=L^{\infty}((t_{0},T);{\mathbb{R}}^{k})$ , then the conclusions of Theorem 19 hold on any bounded set ${\mathcal{B}}\subset L^{\infty}$ containing ${\mathcal{F}}(y_{i})$ . In particular, if (3.30) holds on ${\mathcal{B}}$ , the fixed point $u_{i}$ is the unique global minimizer of $\tfrac{1}{2}\|u\|_{L^{2}}^{2}$ among all $u\in{\mathcal{B}}$ satisfying $L_{u,\tau_{i}}u=y_{i}$ .

Remark 22 (Relation to reachability for fixed $T>t_{0}$ ).

Theorem 19 is fundamentally a synthesis result: given the fixed point $u_{i}\in{\mathcal{F}}(y_{i})$ of ${\mathcal{S}}_{i}$ , it provides an explicit representation of a control that steers $x^{0}$ to $x^{1}$ . It does not assert that such a fixed point exists for every pair $(x^{0},x^{1})$ , nor that the reachable set coincides with ${\mathbb{R}}^{d}$ . Reachability depends on the structure of the feasible coercivity class ${\mathcal{F}}(C_{i})$ and on whether the self-mapping property (3.21) is satisfied. Three distinct situations may occur:

(a) Complete controllability. If there exists $C_{i}>0$ , independent of the initial state $x^{0}\in{\mathbb{R}}^{d}$ , such that ${\mathcal{F}}(C_{i})=L^{\infty}((t_{0},T);{\mathbb{R}}^{k})$ , then (3.21) holds automatically and ${\mathcal{R}}_{\Sigma}(T,x^{0})={\mathbb{R}}^{d}$ for all $x^{0}\in{\mathbb{R}}^{d}$ . In particular, system ( Σ ) is completely controllable on $[t_{0},T]$ . The converse is not true: complete controllability does not imply the existence of $C_{i}>0$ such that ${\mathcal{F}}(C_{i})=L^{\infty}((t_{0},T);{\mathbb{R}}^{k})$ .

(b) Controllability from a fixed $x^{0}$ . If for some $C_{i}=C_{i}(x^{0})>0$ depending on the initial condition $x^{0}$ , one has ${\mathcal{F}}(C_{i})=L^{\infty}((t_{0},T);{\mathbb{R}}^{k})$ , then (3.21) again holds, and ${\mathcal{R}}_{\Sigma}(T,x^{0})={\mathbb{R}}^{d}$ . Thus, the system is controllable from that specific $x^{0}$ , although the property may not extend to all other initial states.

(c) Restricted synthesis. If for some $C_{i}=C_{i}(x^{0})>0$ one only has $\emptyset\neq{\mathcal{F}}(C_{i})\subsetneq L^{\infty}((t_{0},T);{\mathbb{R}}^{k})$ , then Theorem 19 ensures that any fixed point $u_{i}\in{\mathcal{F}}(y_{i})$ steers $x^{0}$ to $x^{1}$ , provided (3.21) is verified. Thus, the synthesis produces a family of feasibly controls, but the reachable set ${\mathcal{R}}_{\Sigma}(T,x^{0})$ may be a proper subset of ${\mathbb{R}}^{d}$ ; its precise description depends on the estimates involving $|\Phi_{T,t_{0}}(x^{1})-x^{0}|$ or $|x^{1}-\Phi_{t_{0},T}(x^{0})|$ that guarantee (3.21).

Directly checking invertibility of ${\mathcal{N}}_{\tau_{i}}(u)$ is generally a difficult task since it depends on the full trajectory $x_{u}(\cdot)$ . This motivates the search for structural assumptions under which, for some $C_{i}>0$ , ${\mathcal{F}}(C_{i})\neq\emptyset$ or coincides with $L^{\infty}((t_{0},T);{\mathbb{R}}^{k})$ . The first such condition is presented below. The proof is presented in Section A.7.

Proposition 23.

Fix $i\in\{1,2\}$ . Suppose $d=k$ and there exists a nonzero $b\in L^{1}((t_{0},T);{\mathbb{R}}_{+})$ such that

|B(t,x)^{\top}y|\ \geq\ b(t)\,|y|\qquad\text{for a.e. }t\in(t_{0},T),\,\forall(x,y)\in{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}.

(3.31)

Then ${\mathcal{N}}_{\tau_{i}}(u):={\mathcal{N}}_{\tau_{i}}(u,x^{0})$ is uniformly coercive, viz.

y^{\top}\,{\mathcal{N}}_{\tau_{i}}(u)\,y\ \geq\ \frac{e^{-2\Lambda_{1}\Delta t_{0}}\,\|b\|_{1}^{2}}{\Delta t_{0}}\,|y|^{2}\qquad\forall(y,\,x^{0},\,u)\in{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}\times L^{\infty}((t_{0},T);{\mathbb{R}}^{k}).

(3.32)

Hence, one may take $C_{i}=\Delta t_{0}\,e^{2\Lambda_{1}\Delta t_{0}}\,\|b\|_{1}^{-2}$ , so that ${\mathcal{F}}(C_{i})=L^{\infty}((t_{0},T);{\mathbb{R}}^{k})$ . In particular, (3.21) is satisfied.

Proposition 23 addresses the fully actuated case $k=d$ under a structural assumption on $B(t,x)$ ensuring uniform coercivity. If this assumption fails, or more generally in the underactuated regime $k<d$ , such global bounds are no longer guaranteed. The following analysis provides sufficient conditions for reachability and synthesis in this configuration. The proof is provided in Section A.8.

Theorem 24.

Let $x^{0}\in{\mathbb{R}}^{d}$ and fix $i\in\{1,2\}$ . Assume the existence of a reference control $u_{i}^{\flat}:=u_{i}^{\flat}(x^{0})\in L^{\infty}((t_{0},T);{\mathbb{R}}^{k})$ such that ${\mathcal{N}}_{\tau_{i}}(u_{i}^{\flat})$ is invertible. Let $C_{i}:=(1+\theta)/\lambda_{\min}\big({\mathcal{N}}_{\tau_{i}}(u_{i}^{\flat})\big)$ where $\theta\in(0,1]$ . Furthermore, assume that the following estimate holds

\|u_{i}^{\flat}\|_{\infty}<\frac{\theta\,\lambda_{\min}({\mathcal{N}}_{\tau_{i}}(u_{i}^{\flat}))}{(1+\theta)\,L_{{\mathcal{N}}_{\tau_{i}}}}.

(3.33)

Then ${\mathcal{F}}(C_{i})\neq\emptyset$ , and (3.21) holds if $y_{i}\in{\mathbb{R}}^{d}$ satisfies

|y_{i}|\leq\vartheta(u_{i}^{\flat})\ :=\ \frac{\lambda_{\min}({\mathcal{N}}_{\tau_{i}}(u_{i}^{\flat}))}{(1+\theta)\,\|B\|_{\infty}\,e^{\Lambda_{1}\Delta t_{0}}}\ \left(\frac{\theta\,\lambda_{\min}({\mathcal{N}}_{\tau_{i}}(u_{i}^{\flat}))}{(1+\theta)\,L_{{\mathcal{N}}_{\tau_{i}}}}-\|u_{i}^{\flat}\|_{\infty}\right).

(3.34)

Here $L_{{\mathcal{N}}_{\tau_{i}}}>0$ is the Lipschitz constant of the map $u\in\{u\in\,L^{\infty}((t_{0},T);{\mathbb{R}}^{k}):\|u\|_{\infty}\leq\,\zeta_{i}\}\mapsto{\mathcal{N}}_{\tau_{i}}(u)$ .

In contrast to Proposition 23, Theorem 24 applies under Assumption 2 alone, requiring no additional structure on $N_{t}$ or $B(t,x)$ . Before discussing its implications, we first establish the Lipschitz constant $L_{{\mathcal{N}}_{\tau_{i}}}>0$ for ${\mathcal{N}}_{\tau_{i}}(\cdot)$ under Assumptions 2 and 5. The proof of the following result is given in Section A.9.

Lemma 25.

Fix $i\in\{1,2\}$ . Then ${\mathcal{N}}_{\tau_{i}}$ defined by (3.10) is continuous from $L^{\infty}((t_{0},T);{\mathbb{R}}^{k})$ to ${\mathcal{S}}_{d}^{+}({\mathbb{R}})$ and globally Lipschitz from ${\mathcal{B}}_{i}:=\{u\in\,L^{\infty}((t_{0},T);{\mathbb{R}}^{k}):\|u\|_{\infty}\leq\,\zeta_{i}\}$ to ${\mathcal{S}}_{d}^{+}({\mathbb{R}})$ . For all $(u,\,v)\in{\mathcal{B}}_{i}^{2}$ ,

$\displaystyle\frac{\\|{\mathcal{N}}_{\tau_{1}}(u)-{\mathcal{N}}_{\tau_{1}}(v)\\|}{\\|u-v\\|_{\infty}}$	$\displaystyle\leq\,L_{{\mathcal{N}}_{\tau_{1}}}:=\frac{L_{B}\\|B\\|_{\infty}^{2}e^{\Lambda_{1}\Delta t_{0}}}{\Lambda_{1}}\Bigg[\frac{e^{(2\Lambda_{1}+L_{B}\zeta_{1})\Delta t_{0}}-e^{\Lambda_{1}\Delta t_{0}}}{\Lambda_{1}+L_{B}\zeta_{1}}+\frac{e^{(2\Lambda_{1}+L_{B}\zeta_{1})\Delta t_{0}}-e^{-\Lambda_{1}\Delta t_{0}}}{L_{B}\zeta_{1}+3\Lambda_{1}}\Bigg]$
	$\displaystyle\quad\qquad+\frac{2\\|B\\|_{\infty}^{3}\Lambda_{2}e^{\Lambda_{1}\Delta t_{0}}}{\Lambda_{1}(\Lambda_{1}+L_{B}\zeta_{1})}\Bigg[\frac{e^{(3\Lambda_{1}+L_{B}\zeta_{1})\Delta t_{0}}-e^{-\Lambda_{1}\Delta t_{0}}}{4\Lambda_{1}+L_{B}\zeta_{1}}-\frac{e^{(2\Lambda_{1}+L_{B}\zeta_{1})\Delta t_{0}}-e^{-\Lambda_{1}\Delta t_{0}}}{3\Lambda_{1}+L_{B}\zeta_{1}}$
	$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+\frac{e^{\Lambda_{1}\Delta t_{0}}-e^{-\Lambda_{1}\Delta t_{0}}}{2\Lambda_{1}}-\frac{e^{2\Lambda_{1}\Delta t_{0}}-e^{-\Lambda_{1}\Delta t_{0}}}{3\Lambda_{1}}\Bigg].$	(3.35)

$\displaystyle\frac{\\|{\mathcal{N}}_{\tau_{2}}(u)-{\mathcal{N}}_{\tau_{2}}(v)\\|}{\\|u-v\\|_{\infty}}$	$\displaystyle\leq\,L_{{\mathcal{N}}_{\tau_{2}}}:=\frac{L_{B}\\|B\\|_{\infty}^{2}e^{\Lambda_{1}\Delta t_{0}}}{\Lambda_{1}}\Bigg[\frac{e^{\Lambda_{1}\Delta t_{0}}-e^{L_{B}\zeta_{2}\Delta t_{0}}}{\Lambda_{1}-L_{B}\zeta_{2}}-\frac{e^{L_{B}\zeta_{2}\Delta t_{0}}-e^{-\Lambda_{1}\Delta t_{0}}}{L_{B}\zeta_{2}+\Lambda_{1}}\Bigg]$
	$\displaystyle\quad\qquad\qquad+\frac{2\\|B\\|_{\infty}^{3}\Lambda_{2}e^{\Lambda_{1}\Delta t_{0}}}{\Lambda_{1}(\Lambda_{1}+L_{B}\zeta_{2})}\Bigg[\frac{e^{2\Lambda_{1}\Delta t_{0}}-e^{L_{B}\zeta_{2}\Delta t_{0}}}{2\Lambda_{1}-L_{B}\zeta_{2}}+\frac{e^{\Lambda_{1}\Delta t_{0}}-e^{L_{B}\zeta_{2}\Delta t_{0}}}{\Lambda_{1}-L_{B}\zeta_{2}}$
	$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\frac{e^{\Lambda_{1}\Delta t_{0}}-e^{-\Lambda_{1}\Delta t_{0}}}{2\Lambda_{1}}-\frac{e^{2\Lambda_{1}\Delta t_{0}}-e^{-\Lambda_{1}\Delta t_{0}}}{3\Lambda_{1}}\Bigg].$	(3.36)

Remark 26.

When $B(t,x)\equiv\,B(t)$ , (25) and (25) holds for $u,\,v\in L^{\infty}((t_{0},T);{\mathbb{R}}^{k})$ , and

L_{{\mathcal{N}}_{\tau_{1}}}:=\frac{\Lambda_{2}\,\|B\|_{\infty}^{3}}{6}\left(3e^{\Lambda_{1}\Delta t_{0}}+1\right)\left(\frac{e^{\Lambda_{1}\Delta t_{0}}-1}{\Lambda_{1}}\right)^{3},\quad\,L_{{\mathcal{N}}_{\tau_{2}}}:=\frac{\Lambda_{2}\,\|B\|_{\infty}^{3}}{3}\left(\frac{e^{\Lambda_{1}\Delta t_{0}}-1}{\Lambda_{1}}\right)^{3}.

The following result is then an immediate consequence.

Corollary 27.

By Lemma 25 and spectral stability

|\lambda_{\min}(C)-\lambda_{\min}(D)|\leq\|C-D\|\qquad\forall(C,\,D)\in{\mathcal{S}}_{d}({\mathbb{R}})^{2},

the map $u\ \mapsto\ \lambda_{\min}\!\big({\mathcal{N}}_{\tau_{i}}(u)\big)$ is continuous on $L^{\infty}((t_{0},T);{\mathbb{R}}^{k})$ and Lipschitz on $\{u\in\,L^{\infty}((t_{0},T);{\mathbb{R}}^{k}):\|u\|_{\infty}\leq\,\zeta_{i}\}$ . In particular, the feasible coercivity class ${\mathcal{F}}(C_{i})$ defined in (3.17) is closed in $L^{\infty}((t_{0},T);{\mathbb{R}}^{k})$ .

Remark 28.

The bounds in (3.34), (25), and (25), while “sharp” in a general sense (under Assumption 2 alone), are conservative for specific vector fields $N_{t}$ . In particular, the exponential factor $e^{\Lambda_{1}\Delta t_{0}}$ reflects the global $\Lambda_{1}$ –Lipschitz bound on $N_{t}$ (cf. Lemma 4), whereas for certain dynamics the differential $\|D\Phi_{t,s}(\cdot)\|$ can in fact decay exponentially. We provide illustrative examples later in Section 3.2.4.

3.2.2 A practical control reference: the zero control

In practice, to check the assumptions of Theorem 24, a natural first choice is $u_{i}^{\flat}\equiv 0=:u_{0}$ , for which the corresponding trajectory reads

x_{u_{0}}(t)=\Phi_{t_{0},t}(x^{0}),\qquad t\in[t_{0},T].

For $i\in\{1,2\}$ and $\tau_{1}=t_{0},\ \tau_{2}=T$ , this yields the symmetric positive semi-definite matrices

W_{i}(T):={\mathcal{N}}_{\tau_{i}}(0,x^{0})=\int_{t_{0}}^{T}\!D\Phi_{t,\tau_{i}}\!\big(\Phi_{t_{0},t}(x^{0})\big)\,B(t,\Phi_{t_{0},t}(x^{0}))B(t,\Phi_{t_{0},t}(x^{0}))^{\top}\,D\Phi_{t,\tau_{i}}\!\big(\Phi_{t_{0},t}(x^{0})\big)^{\top}\,dt.

(3.37)

It is immediate from (2.3) and (2.5) that $W_{2}(T)\in{\mathbb{R}}^{d\times d}$ solves the time-varying Lyapunov differential equation

\dot{W}(t)=B(t,\Phi_{t_{0},t}(x^{0}))B(t,\Phi_{t_{0},t}(x^{0}))^{\top}+DN_{t}\!\big(\Phi_{t_{0},t}(x^{0})\big)\,W(t)+W(t)\,DN_{t}\!\big(\Phi_{t_{0},t}(x^{0})\big)^{\top},\quad W(t_{0})=0.

(3.38)

Moreover, by Remark 9 one has,

W_{2}(t)=S_{t_{0},x^{0}}(t,t_{0})\,W_{1}(t)\,S_{t_{0},x^{0}}(t,t_{0})^{\top},\qquad t\in[t_{0},T]

(3.39)

so $W_{1}(t)$ is invertible iff $W_{2}(t)$ is invertible.

The following result is then an immediate consequence of Theorem 24.

Corollary 29.

Let $x^{0}\in{\mathbb{R}}^{d}$ , $T>t_{0}$ , and fix $i\in\{1,2\}$ . Assume that $W_{2}(T)$ is invertible, and let $C_{i}:=(1+\theta)/\lambda_{\min}(W_{i}(T))$ where $\theta\in(0,1]$ . Then ${\mathcal{F}}(C_{i})\neq\emptyset$ , and (3.21) holds if $y_{i}\in{\mathbb{R}}^{d}$ satisfies

|y_{i}|\leq\frac{\theta\,\lambda_{\min}^{2}(W_{i}(T))}{(1+\theta)^{2}\,L_{{\mathcal{N}}_{\tau_{i}}}\,\|B\|_{\infty}\,e^{\Lambda_{1}\Delta t_{0}}}.

(3.40)

Here $L_{{\mathcal{N}}_{\tau_{i}}}>0$ is the Lipschitz constant of the map $u\in\{u\in\,L^{\infty}((t_{0},T);{\mathbb{R}}^{k}):\|u\|_{\infty}\leq\,\zeta_{i}\}\mapsto{\mathcal{N}}_{\tau_{i}}(u)$ . Furthermore, for any $y_{i}\in{\mathbb{R}}^{d}$ satisfying (3.40), the following energy estimates hold

\int_{t_{0}}^{T}|u_{i}(t)|^{2}\,dt\leq\frac{(1+\theta)}{\lambda_{\min}(W_{i}(T))}|y_{i}|^{2}

(3.41)

where $u_{i}\in{\mathcal{F}}(y_{i})$ is the fixed point of the synthesis map ${\mathcal{S}}_{i}:{\mathcal{F}}(y_{i})\to{\mathcal{F}}(y_{i})$ defined by (3.19).

Proof.

Inequality (3.40) follows from (3.34). Next (3.29) and (3.10) yields

\int_{0}^{T}|u_{i}(t)|^{2}\,dt=\int_{t_{0}}^{T}u_{i}(t)^{\top}\,u_{i}(t)\,dt=y_{i}^{\top}{\mathcal{N}}_{\tau_{i}}(u_{i})^{-1}\,y_{i}\leq\frac{|y_{i}|^{2}}{\lambda_{\min}({\mathcal{N}}_{\tau_{i}}(u_{i}))}\leq\frac{(1+\theta)}{\lambda_{\min}(W_{i}(T))}|y_{i}|^{2},

(3.42)

since $\lambda_{\min}({\mathcal{N}}_{\tau_{i}}(u_{i}))\geq\lambda_{\min}(W_{i}(T))/(1+\theta)$ .

Remark 30.

It follows from Remark (9) that $W_{2}(T)={\mathcal{N}}_{\tau_{2}}(0,x^{0})$ is invertible iff the LTV system

\dot{y}(t)=DN_{t}\!\big(\Phi_{t_{0},t}(x^{0})\big)\,y(t)+B(t,\Phi_{t_{0},t}(x^{0}))\,u(t),\qquad y(t_{0})=y^{0}\in{\mathbb{R}}^{d},\quad t\in[t_{0},T],

(3.43)

is controllable with inputs $u\in L^{1}((t_{0},T);{\mathbb{R}}^{k})$ . This is the Kalman criterion for LTV systems via the controllability Gramian [10, Theorem. 1.16]. Under additional regularity, invertibility of $W_{2}(T)$ can also be characterized by algebraic rank conditions; see, e.g., [7, Theorem 1], [10, Theorem 1.18] and [25, Proposition 3.5.15].

3.2.3 Optimizing the admissible radius: balancing coercivity and control size

The trivial choice $u_{i}^{\flat}=0$ might not be the optimal one in practice to apply Theorem 24 if $W_{2}(T)$ is invertible. In fact, it follows from the admissible target displacement estimate (3.34) that a good reference $u_{i}^{\flat}\in L^{\infty}((t_{0},T);{\mathbb{R}}^{k})$ is one that maximizes $\vartheta(u)$ , which jointly increases the positive coercivity contribution and controls the penalty induced by $\|u\|_{\infty}$ . The aim of this section is to illustrate how we can choose a reference control that maximizes $\vartheta(u)$ .

Let $m\in{\mathbb{N}}$ , $m\geq 1$ , and $\{\varphi_{j}(t):t\in[t_{0},T]\}_{1\leq j\leq m}$ be a chosen (e.g., with respect to the properties of the vector field $N_{t}$ and the input matrix $B$ ) $m$ -linearly independent familly of functions in $L^{\infty}((t_{0},T);{\mathbb{R}}^{k})$ . Fix $i\in\{1,2\}$ , $\theta\in(0,1]$ and introduce the set

{\mathcal{U}}_{\rm ad}:=\left\{u\in\operatorname{Span}\{\varphi_{j}\}:g(u):=\frac{\theta\,\lambda_{\min}({\mathcal{N}}_{\tau_{i}}(u))}{(1+\theta)\,L_{{\mathcal{N}}_{\tau_{i}}}}-\|u\|_{\infty}\geq 0\right\}.

(3.44)

Proposition 31.

Let $x^{0}\in{\mathbb{R}}^{d}$ , $T>t_{0}$ , and fix $i\in\{1,2\}$ . Assume that $W_{2}(T)$ is invertible, and let

\vartheta(u)\ :=\ \frac{\lambda_{\min}({\mathcal{N}}_{\tau_{i}}(u))}{(1+\theta)\,\|B\|_{\infty}\,e^{\Lambda_{1}\Delta t_{0}}}\left(\frac{\theta\,\lambda_{\min}({\mathcal{N}}_{\tau_{i}}(u))}{(1+\theta)\,L_{{\mathcal{N}}_{\tau_{i}}}}-\|u\|_{\infty}\right),\qquad u\in L^{\infty}((t_{0},T);{\mathbb{R}}^{k}).

Then, the following program

\max_{u\in{\mathcal{U}}_{\rm ad}}\ \vartheta(u)

(3.45)

admits at least one solution $u_{i}^{\flat}\in{\mathcal{U}}_{\rm ad}$ . Furthermore, if $\vartheta(u_{i}^{\flat})>0$ then $\lambda_{\min}({\mathcal{N}}_{\tau_{i}}(u_{i}^{\flat}))>0$ , ${\mathcal{F}}(C_{i})\neq\emptyset$ , and (3.21) holds if $y_{i}\in{\mathbb{R}}^{d}$ satisfies $|y_{i}|\leq\vartheta(u_{i}^{\flat})$ where $C_{i}:=(1+\theta)/\lambda_{\min}\big({\mathcal{N}}_{\tau_{i}}(u_{i}^{\flat})\big)$ .

Proof.

Since $W_{2}(T)$ is invertible, $0\in{\mathcal{U}}_{\rm ad}$ so that ${\mathcal{U}}_{\rm ad}\neq\emptyset$ , and ${\mathcal{U}}_{\rm ad}$ is closed since ${\mathcal{U}}_{\rm ad}=g^{-1}([0,\infty))$ and $u\mapsto g(u)$ is continuous. Furthermore, since $\lambda_{\min}({\mathcal{N}}_{\tau_{i}}(u))\leq\|{\mathcal{N}}_{\tau_{i}}(u)\|\leq\|B\|_{\infty}^{2}(e^{2\Lambda_{1}\Delta t_{0}}-1)/(2\Lambda_{1})$ , one deduce that any $u\in{\mathcal{U}}_{\rm ad}$ satisfies

\|u\|_{\infty}\leq\frac{\theta\,\|B\|_{\infty}^{2}}{(1+\theta)\,L_{{\mathcal{N}}_{\tau_{i}}}}\frac{e^{2\Lambda_{1}\Delta t_{0}}-1}{2\Lambda_{1}},

so that ${\mathcal{U}}_{\rm ad}$ is bounded, and therefore compact. Since $u\mapsto\,\vartheta(u)$ is continuous, it follows that (3.45) has at least one solution $u_{i}^{\flat}\in{\mathcal{U}}_{\rm ad}$ by the Weierstrass extreme value theorem. The last part of the proposition follows from Theorem 24, and the following equivalence

\vartheta(u_{i}^{\flat})>0\quad\Longleftrightarrow\quad\left(\|u_{i}^{\flat}\|_{\infty}<\frac{\theta\,\lambda_{\min}({\mathcal{N}}_{\tau_{i}}(u_{i}^{\flat}))}{(1+\theta)\,L_{{\mathcal{N}}_{\tau_{i}}}}\quad\text{and}\quad\lambda_{\min}({\mathcal{N}}_{\tau_{i}}(u_{i}^{\flat}))>0\right).

Remark 32.

In Proposition 31, the assumption that $W_{2}(T)$ is invertible provides the simplest guarantee that ${\mathcal{U}}_{\rm ad}\neq\emptyset$ . More generally, one may only require the existence of $u_{i}^{\ast}\in L^{\infty}((t_{0},T);{\mathbb{R}}^{k})$ such that ${\mathcal{N}}_{\tau_{i}}(u_{i}^{\ast})$ is invertible and $g(u_{i}^{\ast})\geq 0$ . Verifying the latter condition, however, may be nontrivial in concrete models.

3.2.4 Refined estimates for Hopfield-type dynamics

We now return to a key point: the estimates for $\|D\Phi_{t,s}(\cdot)\|$ (Lemma 4) and for $L_{{\mathcal{N}}_{\tau_{i}}}$ (Lemma 25) are somehow “sharp” under Assumption 2, but may be overly conservative for concrete models since they use the global estimates (1.2) and (1.3). We illustrate this fact in this section. Consider the vector field from Hopfield-type recurrent neural networks [15], widely used in theoretical neuroscience and large-scale brain modeling, viz.

N(x)=-D\,x+W\sigma(x),\qquad x\in{\mathbb{R}}^{d},

(3.46)

where $D\in{\mathbb{R}}^{d\times d}$ is diagonal and positive, $W\in{\mathbb{R}}^{d\times d}$ encodes connectivity, and $\sigma$ is the neural activation,

\sigma(x):=(\sigma_{1}(x_{1}),\dots,\sigma_{d}(x_{d}))^{\top},\qquad x:=(x_{1},\,\cdots,\,x_{d})^{\top}\in{\mathbb{R}}^{d},

with $\sigma_{i}(\cdot)$ is of sigmoid-type, say, $\sigma_{i}(s)=\tanh(s)$ . Although $W=W(t)$ may vary analytically in time, e.g. to capture neuron–astrocyte interactions [13], it is commonly assumed to be constant, a simplification that we adopt. As a sigmoid function, $\sigma_{i}$ is infinitely differentiable and globally bounded on ${\mathbb{R}}$ with all its successive derivatives, often $\sigma_{i}^{\prime}(\cdot)\geq 0$ . It follows that $N$ satisfies all the hypotheses of Assumption 2. Since $N$ is autonomous, one assumes that $t_{0}=0$ . In particular, its generated flow $\phi_{t}:=\Phi_{0,t}$ , $t\in{\mathbb{R}}$ , is a one parametric subgroup of $\operatorname{Diff}({\mathbb{R}}^{d})$ , the group of all diffeomorphisms on ${\mathbb{R}}^{d}$ . In this framework, for $i\in\{1,2\}$ , one has

{\mathcal{N}}_{\tau_{i}}(u):={\mathcal{N}}_{\tau_{i}}(u,x^{0})=\int_{0}^{T}\,D\phi_{\tau_{i}-t}(x_{u}(t))\,B(t)\,B(t)^{\top}\,D\phi_{\tau_{i}-t}(x_{u}(t))^{\top}\,dt,\qquad\tau_{1}:=0,\,\tau_{2}:=T,

(3.47)

where the input matrix $B(t,x)\equiv B(t)\in\,L^{\infty}((t_{0},T);{\mathbb{R}}^{d\times k})$ , and $x_{u}$ is the corresponding solution to ( Σ ) with the vector field in (3.46). For ease in notation, we introduce

\|\sigma^{\prime}\|_{\infty}:=\sup_{x\in{\mathbb{R}}^{d}}\|W\,D\sigma(x)\|,\quad\Gamma:=-\lambda_{\min}(D)+\|\sigma^{\prime}\|_{\infty},\quad\Gamma_{1}:=\lambda_{\max}(D)+\|\sigma^{\prime}\|_{\infty},\quad\Gamma_{2}:=\sup_{x\in{\mathbb{R}}^{d}}\|W\,D^{2}\sigma(x)\|.

Then, one has the following result, the proof of which is given in Section A.10.

Proposition 33.

Let $x^{0}\in{\mathbb{R}}^{d}$ and fix $i\in\{1,2\}$ . Then ${\mathcal{N}}_{\tau_{i}}:L^{\infty}((0,T);{\mathbb{R}}^{k})\to{\mathcal{S}}_{d}^{+}({\mathbb{R}})$ is globally Lipschitz,

\|{\mathcal{N}}_{\tau_{1}}(u)-{\mathcal{N}}_{\tau_{1}}(v)\|\leq\frac{\Gamma_{2}\,\|B\|_{\infty}^{3}}{6}\left(3e^{\Gamma_{1}T}+1\right)\left(\frac{e^{\Gamma_{1}T}-1}{\Gamma_{1}}\right)^{3}\,\|u-v\|_{\infty},\qquad\,u,\,v\in\,L^{\infty}((0,T);{\mathbb{R}}^{k}).

(3.48)

\|{\mathcal{N}}_{\tau_{2}}(u)-{\mathcal{N}}_{\tau_{2}}(v)\|\leq\begin{cases}\displaystyle\frac{\Gamma_{2}\,\|B\|_{\infty}^{3}}{3}\left(\frac{e^{\Gamma\,T}-1}{\Gamma}\right)^{3}\,\|u-v\|_{\infty}&\quad\text{if}\quad\Gamma\neq\,0,\cr\cr\displaystyle\frac{\Gamma_{2}\,\|B\|_{\infty}^{3}\,T^{3}}{3}\,\|u-v\|_{\infty}&\quad\text{if}\quad\Gamma=\,0,\end{cases}\qquad\,u,\,v\in\,L^{\infty}((0,T);{\mathbb{R}}^{k}).

(3.49)

Remark 34.

The bounds related to ${\mathcal{N}}_{\tau_{2}}$ may involve a rate $\Gamma\leq 0$ , and are therefore sharper than the general estimates of Remark 26, where a direct application necessarily involves $\Gamma_{1}>0$ as the exponential rate.

Example 35.

Consider the $2$ D Hopfield-type recurrent neural network

\dot{x}=-D\,x+W\,\sigma(x)+Bu,\qquad x(0)=x^{0}\in{\mathbb{R}}^{2}

(3.50)

where $x=(x_{1},x_{2})^{\top}\in{\mathbb{R}}^{2}$ , $u\in L^{\infty}((0,T);{\mathbb{R}})$ , $B=(1,0)^{\top}$ , $D=\operatorname{diag}(d_{1},d_{2})$ with $d_{i}>0$ , $W=(w_{ij})_{1\leq i,j\leq 2}$ , and $\sigma_{i}(s)=\tanh(s)$ . By [26, Theorem 2.1], the necessary and sufficient condition for the complete controllability of (3.50) is that the criterion function

C(\gamma(t))=-d_{2}\gamma_{2}(t)+w_{21}\tanh(\gamma_{1}(t))+w_{22}\tanh(\gamma_{2}(t))

changes its sign over any control curve $\gamma(t):=(\gamma_{1}(t),\gamma_{2}(t))$ solution of

\dot{\gamma}=1,\quad\dot{\gamma}_{2}=0\quad\Longleftrightarrow\quad\gamma_{1}(t)=t+\gamma_{1}^{0},\quad\gamma_{2}(t)=\gamma_{2}^{0},\quad\,t\in{\mathbb{R}},\quad(\gamma_{1}^{0},\gamma_{2}^{0})\in{\mathbb{R}}^{2}.

Since $d_{2}>0$ , and $\tanh$ is bounded, $C(\gamma(t))$ (as a function of $t\in{\mathbb{R}}$ ) does not change its sign for large $\gamma_{2}^{0}$ . Therefore, (3.50) is not completely controllable. It follows that, neither the HCM [8, 22, 29, 16] nor [26, Theorem 2.1] can be directly applied for the motion planning of (3.50). However, one proves using the Kalman-rank condition that ${\mathcal{N}}_{\tau_{i}}(u)$ is invertible for all $(x^{0},u)\in{\mathbb{R}}^{2}\times L^{\infty}((0,T);{\mathbb{R}})$ whenever $w_{21}\neq 0$ . In particular, Corollary 29 applies to (3.50) for all $(x^{0},x^{1})\in{\mathbb{R}}^{2}\times{\mathbb{R}}^{2}$ such that $y_{2}:=x^{1}-\phi_{T}(x^{0})$ satisfies (3.40).

4 Drift–modulated control-affine dynamics with time-varying and state–dependent input matrix

In this section, we consider the more general control-affine system

\dot{x}(t)=A(t,x(t))N_{t}(x(t))+B(t,x(t))u(t),\quad x(t_{0})=x^{0},\quad t\in[t_{0},T].

(4.1)

Throughout the following, $(t,x)\mapsto N_{t}(x)$ satisfies Assumption 2. Whereas, the matrices $A$ and $B$ satisfy the following relaxed regularity assumptions:

Assumption 36.

$A:[t_{0},T]\times{\mathbb{R}}^{d}\to{\mathbb{R}}^{d\times d}$ is an element of $L^{\infty}((t_{0},T)\times{\mathbb{R}}^{d};{\mathbb{R}}^{d\times d})$ , the input matrix $B:[t_{0},T]\times{\mathbb{R}}^{d}\to{\mathbb{R}}^{d\times k}$ is an element of $L^{\infty}((t_{0},T)\times{\mathbb{R}}^{d};{\mathbb{R}}^{d\times k})$ . Additionally, we assume that $x\mapsto A(\cdot,x),\,B(\cdot,x)$ are continuous and locally Lipschitz.

Note that under Assumptions 2 and 36, the Cauchy-Lipschitz theory guarantees the existence of a unique absolutely continuous solution $x\in C^{0}([t_{0},T];{\mathbb{R}}^{d})$ to (4.1) for any control $u\in\,L^{\infty}((t_{0},T);{\mathbb{R}}^{k})$ ; see, for instance, [4, Chapter 2]. Observe also that in contrast to Section 3, here we have only mild smoothness assumptions on $A$ and $B$ . Still, under the trajectory freezing and compactness argument, we illustrate how the control analysis and synthesis of the baseline system ( Σ ) developed in Section 3.2 can be leveraged to provide some insight into the controllability properties of the general control-affine system (4.1).

Fix $z\in C^{0}([t_{0},T];{\mathbb{R}}^{d})$ and define $N_{t}^{z}$ and $B_{z}$ as

N_{t}^{z}(x):=A(t,z(t))N_{t}(x),\quad B_{z}(t):=B(t,z(t))\qquad\forall(t,x)\in[t_{0},T]\times{\mathbb{R}}^{d}.

(4.2)

Then, $(t,x)\mapsto N_{t}^{z}(x)$ satisfies the same regularity assumptions as $(t,x)\mapsto N_{t}(x)$ , as stated in Assumption 2 while $B_{z}$ satisfies Assumption 5. Additionally, the following estimates hold

\|DN_{t}^{z}(w)\|\leq\Lambda_{3}:=\Lambda_{1}\|A\|_{\infty},\quad\|D^{2}N_{t}^{z}(w)\|\leq\Lambda_{4}:=\Lambda_{2}\|A\|_{\infty}\quad\forall(t,w,z)\in[t_{0},T]\times{\mathbb{R}}^{d}\times\,C^{0}([t_{0},T];{\mathbb{R}}^{d})

(4.3)

showing that $w\mapsto N_{t}^{z}(w)$ is globally $\Lambda_{3}$ -Lipschitz continuous on ${\mathbb{R}}^{d}$ , uniformly with respect to $(t,z)$ .

If $t\in[t_{0},T]\to\Phi_{t_{0},t}^{z}$ denotes the nonautonomous flow associated with $N_{t}^{z}$ , then $\{\Phi_{s,t}^{z}\mid(s,t)\in[t_{0},T]\}$ is a two-parameter family of diffeomorphisms which satisfies the same properties as the flow $\Phi_{t_{0},t}$ of $N_{t}$ recalled in Section 2. In particular, we have a similar lemma related to $\Phi_{s,t}^{z}$ as that given in Lemma 4.

Consider the following control-affine system

\dot{x}(t)=N_{t}^{z}(x(t))+B_{z}(t)u(t),\quad x(t_{0})=x^{0},\quad t\in[t_{0},T].

(

\Sigma_{z}

)

As for ( Σ ), system ( Σ z ) has unique absolutely continuous solution $x_{u}^{z}\in C^{0}([t_{0},T];{\mathbb{R}}^{d})$ that can be represented as in Theorem 6 with $b(t,x)=B_{z}(t)u(t)$ . We use the backward representation and get

x_{u}^{z}(t)=\Phi_{T,t}^{z}\!\left(\Phi_{t_{0},T}^{z}(x^{0})+\int_{t_{0}}^{t}D\Phi_{s,T}^{z}\big(x_{u}^{z}(s)\big)\,B_{z}(s)\,u(s)\,ds\right),\qquad t\in[t_{0},T].

(4.4)

Fix $x^{0}\in{\mathbb{R}}^{d}$ , $u\in L^{\infty}((t_{0},T);{\mathbb{R}}^{k})$ and let $x_{u}^{z}:=x_{u,x^{0}}^{z}\in C^{0}([t_{0},T];{\mathbb{R}}^{d})$ be the corresponding solution of ( Σ z ). We define ${\mathcal{N}}_{\tau_{2}}^{z}(u):={\mathcal{N}}_{\tau_{2}}^{z}(u,x^{0})$ similarly as was defined ${\mathcal{N}}_{\tau_{2}}(u)$ in Section 3.2, viz.

{\mathcal{N}}_{\tau_{2}}^{z}(u)=\int_{t_{0}}^{T}D\Phi_{t,T}^{z}(x_{u}^{z}(t))B_{z}(t)B_{z}(t)^{\top}D\Phi_{t,T}^{z}(x_{u}^{z}(t))^{\top}\,dt,

(4.5)

which is a symmetric positive-semidefinite matrix. In particular, Remark 26 yields

\|{\mathcal{N}}_{\tau_{2}}^{z}(u)-{\mathcal{N}}_{\tau_{2}}^{z}(v)\|\leq\frac{\Lambda_{4}\,\|B\|_{\infty}^{3}}{3}\left(\frac{e^{\Lambda_{3}\Delta t_{0}}-1}{\Lambda_{3}}\right)^{3}\,\|u-v\|_{\infty}\qquad\forall\,u,\,v\in\,L^{\infty}((t_{0},T);{\mathbb{R}}^{k}),

(4.6)

showing that ${\mathcal{N}}_{\tau_{2}}^{z}:L^{\infty}((t_{0},T);{\mathbb{R}}^{k})\to{\mathcal{S}}_{d}^{+}({\mathbb{R}})$ is uniformly globally Lipschitz w.r.t. $z\in C^{0}([t_{0},T];{\mathbb{R}}^{d})$ .

Similarly, as for $W_{2}(T)$ defined in (3.37), one introduces the matrix

W_{2}^{z}(T):={\mathcal{N}}_{\tau_{2}}^{z}(0)=\int_{t_{0}}^{T}D\Phi_{t,T}^{z}(\Phi_{t_{0},t}^{z}(x^{0}))B_{z}(t)B_{z}(t)^{\top}D\Phi_{t,T}^{z}(\Phi_{t_{0},t}^{z}(x^{0}))^{\top}\,dt,

(4.7)

which is symmetric positive semi-definite.

The main theorem of this section is then the following.

Theorem 37.

Let $x^{0}\in{\mathbb{R}}^{d}$ . Assume that there exists $C>0$ such that

\lambda_{\min}(W_{2}^{z}(T))\geq C^{-1}\qquad\forall z\in C^{0}([t_{0},T];{\mathbb{R}}^{d}),

(4.8)

and that, for each frozen system, the corresponding synthesis map satisfies the contractivity condition of Theorem 14. Then, there exist $z_{*}\in C^{0}([t_{0},T];{\mathbb{R}}^{d})$ such that (4.1) is controllable on $[t_{0},T]$ from $x^{0}$ to any target state $x^{1}\in{\mathbb{R}}^{d}$ satisfying for some $\theta\in(0,1)$ the following estimate

|x^{1}-\Phi_{t_{0},T}^{z_{*}}(x^{0})|\leq\frac{\theta\,\lambda_{\min}\big(W_{2}^{z_{*}}(T)\big)}{(1+\theta)\,L_{*}\,\|B\|_{\infty}\,e^{\Lambda_{3}\Delta t_{0}}}.

(4.9)

Here $L_{*}>0$ is the Lipschitz constant in (4.6).

Remark 38.

Following Theorem 24, the assumption in Theorem 37 can be relaxed as follows: assume the existence of a reference control $u_{2}^{\flat}\in L^{\infty}((t_{0},T);{\mathbb{R}}^{k})$ and a constant $C>0$ such that

\lambda_{\min}\big({\mathcal{N}}_{\tau_{2}}^{z}(u_{2}^{\flat})\big)\ \geq\ C^{-1},\qquad\forall\,z\in C^{0}([t_{0},T];{\mathbb{R}}^{d}).

Moreover, as illustrated in Section 3.2.4, the constant $L_{\ast}>0$ is often conservative, and can be sharpened in concrete models by exploiting structural properties of the vector field $N_{t}^{z}$ .

Proof of Theorem 37.

Let $z\in C^{0}([t_{0},T];{\mathbb{R}}^{d})$ . By (4.8), $W_{2}^{z}(T)$ is invertible. Then, letting $C_{2}:=(1+\theta)/C$ with $C>0$ as in (4.8), and for some $\theta\in(0,1]$ , the feasible coercivity class

{\mathcal{F}}(C_{2})=\{u\in L^{\infty}((t_{0},T);{\mathbb{R}}^{k}):\lambda_{\min}({\mathcal{N}}_{\tau_{2}}^{z}(u))\geq C_{2}^{-1}\}

(4.10)

is nonempty since it contains $u=0$ . For $x^{1}\in{\mathbb{R}}^{d}$ , if the following estimate holds

|x^{1}-\Phi_{t_{0},T}^{z}(x^{0})|\leq\frac{\theta\,\lambda_{\min}\big(W_{2}^{z}(T)\big)}{(1+\theta)\,L_{*}\,\|B\|_{\infty}\,e^{\Lambda_{3}\Delta t_{0}}},

(4.11)

then by Remark 26, and Theorems 14 and 19, the fixed point $u_{z}\in{\mathcal{F}}(C_{2})$ of

{\mathcal{S}}_{2}^{z}(u)(t)=B_{z}(t)^{\top}D\Phi_{t,T}^{z}(x_{u}(t))^{\top}({\mathcal{N}}_{\tau_{2}}^{z}(u))^{-1}\left(x^{1}-\Phi_{t_{0},T}^{z}(x^{0})\right)

exists, is unique, and the corresponding solution $x_{u_{z}}$ to system ( Σ z ) satisfies $x_{u_{z}}(T)=x^{1}$ .

To complete the proof of the theorem, let us define the map

{\mathcal{Z}}:z\in C^{0}([t_{0},T];{\mathbb{R}}^{d})\mapsto{\mathcal{Z}}(z)=x_{u_{z}}\in C^{0}([t_{0},T];{\mathbb{R}}^{d})

(4.12)

where $x_{u_{z}}\in C^{0}([t_{0},T];{\mathbb{R}}^{d})$ is the solution of ( Σ z ) corresponding to the control $u_{z}={\mathcal{S}}_{2}^{z}(u_{z})$ .

If ${\mathcal{Z}}$ has a fixed point $z_{*}\in C^{0}([t_{0},T];{\mathbb{R}}^{d})$ , then $z_{*}=x_{u_{z_{*}}}$ , and by construction, $u_{z_{*}}$ steers (4.1) from $x^{0}$ to any target state $x^{1}\in{\mathbb{R}}^{d}$ satisfying (4.9), i.e., $x_{u_{z_{*}}}$ is the solution to system (4.1) and satisfies $x_{u_{z_{*}}}(T)=x^{1}$ .

First, ${\mathcal{Z}}$ is clearly well-defined and continuous. Moreover, by Lemma 4 and (4.8), it holds that

\|u_{z}\|_{\infty}\leq C\|B\|_{\infty}e^{\Lambda_{3}\Delta t_{0}}(e^{\Lambda_{3}\Delta t_{0}}|x^{1}|+|x^{0}|)\quad\forall z\in C^{0}([t_{0},T];{\mathbb{R}}^{d})

(4.13)

so that for some $M>0$ independent of $z$ , we find

\|{\mathcal{Z}}(z)\|_{\infty}=\|x_{u_{z}}\|_{\infty}\leq e^{\Lambda_{1}\Delta t_{0}}(|x^{0}|+\Delta t_{0}(\|B\|_{\infty}\|u_{z}\|_{\infty}))\leq M\quad\forall z\in C^{0}([t_{0},T];{\mathbb{R}}^{d}).

(4.14)

Next, ${\mathcal{Z}}$ is compact. In fact, let ${\mathcal{B}}\subset C^{0}([t_{0},T];{\mathbb{R}}^{d})$ be a bounded set, let us show that $\overline{{\mathcal{Z}}({\mathcal{B}})}$ is compact. First, ${\mathcal{Z}}({\mathcal{B}})\subset C^{0}([t_{0},T];{\mathbb{R}}^{d})$ is bounded by (4.14). It remains to prove that for all $t_{*}\in[t_{0},T]$ , the following holds

\sup\limits_{z\in{\mathcal{B}}}|{\mathcal{Z}}(z)(t)-{\mathcal{Z}}(z)(t_{*})|\xrightarrow[t\to t_{*}]{}0.

(4.15)

One has immediately from (4.13) and (4.14) that for some $M_{1}>0$ , it holds that

|{\mathcal{Z}}(z)(t)-{\mathcal{Z}}(z)(t_{*})|=|x_{u_{z}}(t)-x_{u_{z}}(t_{*})|\leq(\Lambda_{3}\|x_{u_{z}}\|_{\infty}+\|B\|_{\infty}\|u_{z}\|_{\infty})|t-t_{*}|\leq M_{1}|t-t_{*}|

(4.16)

for every $t\in[t_{0},T]$ . This proves (4.15) and, therefore, that ${\mathcal{Z}}$ is compact by the Ascoli-Arzelà theorem. Finally, the set $\{z\in C^{0}([t_{0},T];{\mathbb{R}}^{d})\mid\,z=\theta{\mathcal{Z}}(z)\,\text{for some}\,\theta\in[0,1]\}$ is bounded by (4.13) and (4.14). According to Schaefer’s fixed point theorem, ${\mathcal{Z}}$ admits at least one fixed point $z_{*}\in C^{0}([t_{0},T];{\mathbb{R}}^{d})$ , i.e., $z_{*}={\mathcal{F}}(z_{*})=x_{u_{z_{*}}}$ .

Remark 39.

Theorem 37 generalizes [10, Theorem 3.40] beyond the trivial case $N_{t}(x)=x$ . In this setting, $N_{t}^{z}(x)=A(t,z(t))x$ , so ${\mathcal{N}}_{\tau_{2}}^{z}(u)$ is independent of $u$ and the admissible class in (4.10) reduces to ${\mathcal{F}}(C_{2})=L^{\infty}((t_{0},T);{\mathbb{R}}^{k})$ for all initial states and trajectories by letting $C_{2}:=C$ with $C>0$ as in (4.8). Hence, by arguments analogous to Proposition 23, the baseline system ( Σ z ) is globally controllable for all $z$ , and global controllability of (4.1) follows via Schaefer’s fixed point theorem.

5 Conclusion

We developed a constructive framework of control synthesis for control–affine system $\dot{x}=N_{t}(x)+B(t,x)u$ based on trajectory–dependent Gramians and fixed–point synthesis maps. On a ball ${\mathcal{F}}(y_{i})$ of the feasible coercivity class ${\mathcal{F}}(C_{i})$ , sufficient conditions ensure that the synthesis map is a self–map and admits a unique fixed point with the associated control satisfying an explicit energy certificate. Beyond synthesis, we established existence of minimizers: the feasible set $\mathfrak{F}_{i}=\{u\in{\mathcal{F}}(y_{i}):\,L_{u,\tau_{i}}u=y_{i}\}$ is weakly sequentially closed in $L^{2}((t_{0},T);{\mathbb{R}}^{k})$ , hence $\tfrac{1}{2}\|u\|_{L^{2}}^{2}$ attains a minimum on $\mathfrak{F}_{i}$ (Proposition 18). We then linked optimality to fixed points: under the orthogonality condition at the fixed point (Theorem 17) and a finite-dimensional invertibility assumption, every local minimizer over $\mathfrak{F}_{i}$ equals the fixed point $u_{i}$ ; therefore, the minimizer on $\mathfrak{F}_{i}$ is unique and global. If, in addition, ${\mathcal{F}}(C_{i})=L^{\infty}((t_{0},T);{\mathbb{R}}^{k})$ , the same conclusion holds among all bounded controls satisfying the endpoint constraint.

The framework extends the classical Gramian theory for linear systems and, for (1.1), recovers [10, Theorem. 3.40] in the case of $N_{t}(x)=x$ . Moreover, our results apply to Hopfield–type networks [15] and Lur’e systems [11] in general. It is constructive and amenable to numerical implementation. The analysis is pointwise in the initial state; natural next steps include deriving uniform state–space conditions paralleling the linear equivalence between controllability and Gramian invertibility, investigating robustness, and obtaining a sharper quantitative characterization of the reachable set.

When complete controllability on $[t_{0},T]$ is known a priori for system $\dot{x}=N_{t}(x)+B(t,x)u$ , our framework suggests a systematic procedure: certify on short windows that (i) ${\mathcal{F}}(C_{i})$ is nonempty (via an invertible reference Gramian), and (ii) the synthesis map is a self–map and satisfies an iterate-contractivity condition; then concatenate the windowed fixed points along a partition of $[t_{0},T]$ to steer arbitrary targets, with additive energy bounds. This “windowed Gramian synthesis” provides an implementable alternative to homotopy continuation, equipped with an energy certificate and an a posteriori optimality upgrade at the concatenated fixed point via the orthogonality condition test. Establishing uniform window conditions from structural data and extending the patching argument to underactuated regimes are promising directions for future work.

Appendix A Proofs of results stated in the main text

In this section, we present proofs of some of the results from the main text. We start with the following.

A.1 Proof of Theorem 6

We present in this section the proof of the solution representation (3.2).

Proof.

Under the assumptions of Theorem 6, system

\dot{x}(t)=N_{t}(x(t))+b(t,x(t)),\qquad x(t_{0})=x^{0}

(A.1)

has (see, for instance, [4, Chapter 2]) a unique absolutely continuous solution $x\in C^{0}([t_{0},T];{\mathbb{R}}^{d})$ . Let us prove that this solution can be represented by (3.2). Let $t\in[t_{0},T]\mapsto y(t)\in{\mathbb{R}}^{d}$ be such that $t\mapsto x(t)=\Phi_{\tau_{i},t}(y(t))$ is the solution of (A.1). Then $y$ is derivable almost everywhere w.r.t. $t$ and it holds that

N_{t}(x(t))+b(t,x(t))=\dot{x}(t)=N_{t}(x(t))+D\Phi_{\tau_{i},t}(y(t))\,\dot{y}(t)

(A.2)

so that using (2.6), we find that $y$ solves the following.

\dot{y}(t)=D\Phi_{t,\tau_{i}}(x(t))\,b(t,x(t)),\qquad y(t_{0})=\Phi_{t_{0},\tau_{i}}(x^{0}).

(A.3)

Integrating (A.3) yields (3.2). Conversely, if (3.2) holds, then $x(t_{0})=\Phi_{\tau_{i},t_{0}}(\Phi_{t_{0},\tau_{i}}(x^{0}))=x^{0}$ and $x\in C^{0}([t_{0},T];{\mathbb{R}}^{d})$ by composition. Otherwise, there exists $(t_{n})\subset[t_{0},T]$ , $t_{*}\in[t_{0},T]$ with $t_{n}\to t_{*}$ and $\varepsilon>0$ such that $|\Phi_{t,\tau_{i}}(x(t_{n}))-\Phi_{t_{*},\tau_{i}}(x(t_{*}))|\geq\varepsilon$ . In fact, $x\in C^{0}([t_{0},T];{\mathbb{R}}^{d})$ if and only if $t\mapsto y(t):=\Phi_{t,\tau_{i}}(x(t))$ belongs to $C^{0}([t_{0},T];{\mathbb{R}}^{d})$ since $\Phi_{t,\tau_{i}}$ is invertible and $C^{1}$ w.r.t. $t\in{\mathbb{R}}$ . However, we have from (3.2) that

\Phi_{t_{n},\tau_{i}}(x(t_{n}))-\Phi_{t_{*},\tau_{i}}(x(t_{*}))=\int_{t_{*}}^{t_{n}}D\Phi_{s,\tau_{i}}(x(s))\,b(s,x(s))\,ds

which implies $|\Phi_{t_{n},\tau_{i}}(x(t_{n}))-\Phi_{t_{*},\tau_{i}}(x(t_{*}))|\leq\|b\|_{\infty}e^{\Lambda_{1}|\tau_{i}-t_{*}|}|t_{n}-t_{*}|$ by using (2.7) with $\beta(s)=x(s)$ . It follows that $|\Phi_{t_{n},\tau_{i}}(x_{u}(t_{n}))-\Phi_{t_{*},\tau_{i}}(x_{u}(t_{*}))|\to 0$ as $n\to\infty$ , which is inconsistent. Letting now

z(t):=\Phi_{t_{0},\tau_{i}}(x^{0})+\int_{t_{0}}^{t}D\Phi_{s,\tau_{i}}(x(s))\,b(s,x(s))\,ds

(A.4)

and deriving (3.2) almost everywhere w.r.t. $t$ yields

\dot{x}(t)=\partial_{t}\Phi_{\tau_{i},t}(z(t))+D\Phi_{\tau_{i},t}(z(t))\,D\Phi_{t,\tau_{i}}(x(t))\,b(t,x(t))=N_{t}(\Phi_{\tau_{i},t}(z(t)))+b(t,x(t))=N_{t}(x(t))+b(t,x(t))

(A.5)

by $x(t)=\Phi_{\tau_{i},t}(z(t))$ and $D\Phi_{\tau_{i},t}(z(t))\,D\Phi_{t,\tau_{i}}(x(t))={\operatorname{Id}}$ . It follows that (3.2) solves (A.1).

A.2 Proof of Lemma 8

Proof.

Recall from [4, Theorem 3.2.6] that the Fréchet derivative $D_{u}x_{u}(t)h$ of $x_{u}(t)$ w.r.t. $u$ in the direction of $h\in L^{\infty}((t_{0},T);{\mathbb{R}}^{k})$ is given by

D_{u}x_{u}(t)h=\int_{t_{0}}^{t}\!\!\!R_{u}(t,s)B(s,x_{u}(s))h(s)\,ds.

(A.6)

Using the solution representation (3.2) with $b(t,x(t))=B(t,x(t))u(t)$ and letting $y_{u}(t):=\Phi_{t,\tau_{i}}(x_{u}(t))$ , one finds $D_{u}x_{u}(t)h=\big[D\Phi_{t,\tau_{i}}(x_{u}(t))\big]^{-1}D_{u}y_{u}(t)h$ and

\frac{d}{dt}D_{u}y_{u}(t)h={\mathcal{A}}_{\tau_{i},u}(t)D_{u}y_{u}(t)h+D\Phi_{t,\tau_{i}}(x_{u}(t))B(t,x_{u}(t))h(t),\qquad D_{u}y_{u}(t)h|_{t=t_{0}}=0

where ${\mathcal{A}}_{\tau_{i},u}(t)$ is defined as in the statement of Lemma 8. It follows that

D_{u}y_{u}(t)h=\int_{t_{0}}^{t}\!\!M_{\tau_{i},u}(t,s)D\Phi_{s,\tau_{i}}(x_{u}(s))B(s,x_{u}(s))h(s)\,ds

(A.7)

where $M_{\tau_{i},u}(t,s)$ is defined as in the statement of Lemma 8. One deduces that

D_{u}x_{u}(t)h=\big[D\Phi_{t,\tau_{i}}(x_{u}(t))\big]^{-1}\int_{t_{0}}^{t}\!\!M_{\tau_{i},u}(t,s)D\Phi_{s,\tau_{i}}(x_{u}(s))B(s,x_{u}(s))h(s)\,ds.

(A.8)

Next, (A.6) and (A.8) suggest the factorization $R_{u}(t,s)=\big[D\Phi_{t,\tau_{i}}(x_{u}(t))\big]^{-1}M_{\tau_{i},u}(t,s)D\Phi_{s,\tau_{i}}(x_{u}(s))$ . Using this factorization, one checks immediately that

R_{u}(t_{1},t_{1})={\operatorname{Id}},\quad R_{u}(t_{1},t_{2})R_{u}(t_{2},t_{3})=R_{u}(t_{1},t_{3}),\quad R_{u}(t_{1},t_{2})R_{u}(t_{2},t_{1})={\operatorname{Id}},\qquad\forall(t_{1},t_{2},t_{3})\in[t_{0},T]^{3}

since $M_{\tau_{i},u}(t,s)$ is a state-transition matrix. Finally, using the following identities

	$\displaystyle\frac{d}{dt}\big[D\Phi_{t,\tau_{i}}(x_{u}(t))\big]^{-1}$	$\displaystyle=$	$\displaystyle-\big[D\Phi_{t,\tau_{i}}(x_{u}(t))\big]^{-1}\left(\frac{d}{dt}D\Phi_{t,\tau_{i}}(x_{u}(t))\right)\big[D\Phi_{t,\tau_{i}}(x_{u}(t))\big]^{-1}$		(A.9)
		$\displaystyle=$	$\displaystyle DN_{t}(x_{u}(t))\big[D\Phi_{t,\tau_{i}}(x_{u}(t))\big]^{-1}+D^{2}\Phi_{\tau_{i},t}\big(\Phi_{t,\tau_{i}}(x_{u}(t))\big)D\Phi_{t,\tau_{i}}(x_{u}(t))B(t,x_{u}(t))u(t),$		(A.9)

D^{2}\Phi_{\tau_{i},t}\big(\Phi_{t,\tau_{i}}(x_{u}(t))\big)D\Phi_{t,\tau_{i}}(x_{u}(t))D\Phi_{t,\tau_{i}}(x_{u}(t))+D\Phi_{\tau_{i},t}\big(\Phi_{t,\tau_{i}}(x_{u}(t))\big)D^{2}\Phi_{t,\tau_{i}}(x_{u}(t))=0,

(A.10)

and the fact that $M_{\tau_{i},u}(t,s)$ is a state-transition matrix of (3.7), one finds that

\frac{\partial R_{u}}{\partial t}(t,s)=\big[DN_{t}(x_{u}(t))+D_{x}B(t,x_{u}(t))u(t)\big]R_{u}(t,s),\qquad\quad(t,s)\in[t_{0},T]^{2}.

(A.11)

This completes the proof of the Lemma.

A.3 Proof of Theorem 14

In this section, we present the proof of Theorem 14 that we split into several steps for the reader’s convenience. Since the maps ${\mathcal{S}}_{1}$ and ${\mathcal{S}}_{2}$ play similar roles, we only focus on ${\mathcal{S}}_{2}$ . Throughout this section, we set ${\mathcal{Y}}:=L^{\infty}((t_{0},T);{\mathbb{R}}^{k})$ . Under the assumptions of Theorem 14,

{\mathcal{S}}_{2}(u)=B_{u}(t)^{\top}\,Q_{u}(t)^{\top}\,{\mathcal{N}}_{\tau_{2}}(u)^{-1}\,y_{2}\qquad\text{and}\qquad{\mathcal{S}}_{2}(u)\in{\mathcal{F}}(y_{2})\qquad\forall u\in{\mathcal{F}}(y_{2}),

where $y_{2}:=x^{1}-\Phi_{t_{0},T}(x^{0})$ , $B_{u}(t):=B(t,x_{u}(t))$ and $Q_{u}(t):=D\Phi_{t,T}(x_{u}(t))$ . For ease in notation, for $s\in[0,\Delta t_{0}]$ , where $\Delta t_{0}:=T-t_{0}$ , we let

E_{0}=e^{L_{B}\,\zeta_{2}\,\Delta t_{0}},\;\;E_{1}(s)=e^{\Lambda_{1}s},\;\,E_{2}(s)=e^{2\Lambda_{1}s}-e^{\Lambda_{1}s},\;\,E_{1,\infty}=\max_{s\in[0,\Delta t_{0}]}E_{1}(s),\;\,E_{2,\infty}=\max_{s\in[0,\Delta t_{0}]}E_{2}(s).

(A.12)

We introduce the following Volterra and Fredholm-type bounded linear operators $V,\;F:{\mathcal{Y}}\to{\mathcal{Y}}$ defined by

(Vf)(t)=\int_{t_{0}}^{t}f(\tau)\,d\tau,\qquad Ff=\int_{t_{0}}^{T}f(\tau)\,d\tau\qquad\forall(t,\,f)\in[t_{0},T]\times{\mathcal{Y}}.

(A.13)

Lemma 40.

For every $(u,\,v)\in{\mathcal{F}}(y_{2})^{2}$ , the following estimates hold

|x_{u}(t)-x_{v}(t)|\leq\|B\|_{\infty}\,E_{0}\,E_{1,\infty}\,(V|u-v|)(t)\leq\|B\|_{\infty}\,E_{0}\,E_{1,\infty}\,\Delta t_{0}\|u-v\|_{\infty}\qquad\forall t\in[t_{0},T],

(A.14)

\|{\mathcal{N}}_{\tau_{2}}(u)-{\mathcal{N}}_{\tau_{2}}(v)\|\leq 2\,E_{0}\,E_{1,\infty}\,\|B\|_{\infty}^{2}\bigg(\frac{\|B\|_{\infty}\Lambda_{2}\,E_{2,\infty}}{\Lambda_{1}}+L_{B}\,E_{1,\infty}\bigg)\,F\circ V|u-v|.

(A.15)

Proof.

Recall that $\dot{x}_{u}(t)=N_{t}(x_{u}(t))+B(t,x_{u}(t))u(t)$ , $x_{u}(t_{0})=x^{0}$ and $\dot{x}_{v}(t)=N_{t}(x_{v}(t))+B(t,x_{u}(t))v(t)$ , $x_{v}(t_{0})=x^{0}$ . Then, by the Cauchy-Schwarz inequality, one finds that

\frac{d}{dt}|x_{u}(t)-x_{v}(t)|=\frac{\langle\dot{x}_{u}(t)-\dot{x}_{v}(t),x_{u}(t)-x_{v}(t)\rangle}{|x_{u}(t)-x_{v}(t)|}\leq(\Lambda_{1}+L_{B}\,\zeta_{2})\,|x_{u}(t)-x_{v}(t)|+\|B\|_{\infty}\,|u(t)-v(t)|

which by Gronwall’s lemma yields (A.14). Now using Lemma 4 respectively with $\beta\in\{x_{u},x_{v}\}$ , one finds

\|{\mathcal{N}}_{\tau_{2}}(u)-{\mathcal{N}}_{\tau_{2}}(v)\|\leq\,2\bigg(\frac{\|B\|_{\infty}^{2}\Lambda_{2}\,E_{2,\infty}}{\Lambda_{1}}+L_{B}\|B\|_{\infty}\,E_{1,\infty}\bigg)\int_{t_{0}}^{T}\,|x_{u}(s)-x_{v}(s)|\,ds

(A.16)

which by the first inequality in (A.14) yields (A.15).

We introduce the positive constants

\alpha_{1}=2E_{0}C_{2}^{2}\|B\|_{\infty}^{3}E_{1,\infty}^{2}\bigg(\frac{\Lambda_{2}\|B\|_{\infty}E_{2,\infty}}{\Lambda_{1}}+L_{B}E_{1,\infty}\bigg)|y_{2}|,\;\alpha_{2}=E_{0}C_{2}\|B\|_{\infty}E_{1,\infty}\bigg(\frac{\Lambda_{2}\|B\|_{\infty}E_{2,\infty}}{\Lambda_{1}}+L_{B}E_{1,\infty}\bigg)|y_{2}|,

(A.17)

and the bounded and linear operator ${\mathcal{L}}:{\mathcal{Y}}\to{\mathcal{Y}}$ , defined by

({\mathcal{L}}f)(t)=\alpha_{1}\,F\circ Vf+\alpha_{2}(Vf)(t)\qquad\forall(t,\,f)\in[t_{0},T]\times{\mathcal{Y}}.

(A.18)

Lemma 41.

Set

K_{2}:=\Delta t_{0}(\alpha_{1}\Delta t_{0}+\alpha_{2}),\qquad p_{2}:=\frac{\alpha_{1}\Delta t_{0}}{\alpha_{1}\Delta t_{0}+\alpha_{2}}\in(0,1).

(A.19)

Let $\gamma_{m}(p_{2}):=z_{m}(1)$ , where $z_{0}\equiv 1$ on $[0,1]$ and

z_{m+1}(y)=(1-p_{2})\int_{0}^{y}z_{m}(r)\,dr+p_{2}\int_{0}^{1}(1-r)z_{m}(r)\,dr.

(A.20)

Then, for every $m\geq 1$ ,

\|{\mathcal{L}}^{m}\|\leq K_{2}^{m}\gamma_{m}(p_{2}).

(A.21)

Proof.

Introduce the isometric rescaling $\mathcal{R}f(y):=f(t_{0}+\Delta t_{0}y)$ on $L^{\infty}(0,1)$ , and let $(Wg)(y)=\int_{0}^{y}g(r)dr$ and $Jg=\int_{0}^{1}g(r)dr$ . Then $\mathcal{R}V\mathcal{R}^{-1}=\Delta t_{0}W$ and $\mathcal{R}F\mathcal{R}^{-1}=\Delta t_{0}J$ . Hence $\widetilde{{\mathcal{L}}}:=\mathcal{R}{\mathcal{L}}\mathcal{R}^{-1}=K_{2}T_{p_{2}}$ , with $T_{p_{2}}:=(1-p_{2})W+p_{2}JW$ .

Since $T_{p_{2}}$ is positive, for $\|g\|_{\infty}\leq 1$ one has $|T_{p_{2}}^{m}g|\leq T_{p_{2}}^{m}\mathbf{1}$ a.e. Let $z_{m}:=T_{p_{2}}^{m}\mathbf{1}$ . Then $z_{0}\equiv 1$ and $z_{m+1}=T_{p_{2}}z_{m}$ , which gives exactly (A.20). The functions $z_{m}$ are nonnegative and nondecreasing by induction; hence $\|z_{m}\|_{\infty}=z_{m}(1)=\gamma_{m}(p_{2})$ . Thus $\|T_{p_{2}}^{m}\|\leq\gamma_{m}(p_{2})$ , and since $\mathcal{R}$ is an isometry, $\|{\mathcal{L}}^{m}\|=\|\widetilde{{\mathcal{L}}}^{\,m}\|\leq K_{2}^{m}\gamma_{m}(p_{2})$ .

Lemma 42.

Let $p\in(0,1)$ , and let $\lambda(p)>0$ be the unique positive solution of

p(e^{\lambda}-1)-\lambda=0.

(A.22)

Set $\Lambda(p):=\lambda(p)/(1-p)$ . Then $\Lambda(p)>2$ . Moreover, if $0<K<\Lambda(p)$ , then $K^{m}\gamma_{m}(p)\to 0$ as $m\to\infty$ .

Proof.

For $T_{p}=(1-p)W+pJW$ , the same argument as in the proof of Lemma 41 gives $\|T_{p}^{m}\|\leq\gamma_{m}(p)$ , while testing on $\mathbf{1}$ gives the reverse inequality. Hence $\gamma_{m}(p)=\|T_{p}^{m}\|$ . The compact operator $T_{p}$ is strongly positive on the cone of nonnegative functions in $L^{\infty}(0,1)$ , so the Krein–Rutman theorem [5, Theorem 6.13 & Problem 41] gives an eigenpair $\mu=r(T_{p})>0$ and $f\in\operatorname{Int}P$ such that $T_{p}f=\mu f$ . It follows that $f$ has an absolutely continuous representation on $[0,1]$ that we still denote by $f$ . Differentiating $T_{p}f=\mu f$ gives $\mu f^{\prime}=(1-p)f$ , hence $f(y)=f(0)e^{\lambda y}$ with $\lambda=(1-p)/\mu$ . Evaluating at $y=0$ gives (A.22); hence $r(T_{p})=1/\Lambda(p)$ . The inequality $\Lambda(p)>2$ follows by setting $q=1-p$ and observing that the defining function is negative at $\lambda=2q$ , equivalently $2q<\log((1+q)/(1-q))$ . The spectral-radius formula gives $\lim_{m\to\infty}\gamma_{m}(p)^{1/m}=1/\Lambda(p)$ , and therefore $K^{m}\gamma_{m}(p)\to 0$ whenever $K<\Lambda(p)$ .

Proof of Theorem 14.

Let $(u,\,v)\in{\mathcal{F}}(y_{2})^{2}$ . Then, for every $t\in[t_{0},T]$ , it hols that

	$\displaystyle\|{\mathcal{S}}_{2}(u)(t)-{\mathcal{S}}_{2}(v)(t)\|$	$\displaystyle\leq$	$\displaystyle\|y_{2}\|C_{2}\big(\\|B\\|_{\infty}\,E_{1,\infty}\,C_{2}\\|{\mathcal{N}}_{\tau_{2}}(u)-{\mathcal{N}}_{\tau_{2}}(v)\\|+\\|B\\|_{\infty}\\|Q_{u}(t)-Q_{v}(t)\\|+E_{1,\infty}\\|B_{u}(t)-B_{v}(t)\\|\big)$		(A.23)
		$\displaystyle\leq$	$\displaystyle\alpha_{1}\,F\circ V\|u-v\|+\alpha_{2}(V\|u-v\|)(t)=({\mathcal{L}}\|u-v\|)(t)$		(A.23)

by Lemma 4, (A.14), (A.15), (A.17) and (A.18). It follows that for every $m\in{\mathbb{N}}$ , with $m\geq 1$ , it holds that

|{\mathcal{S}}_{2}^{m}(u)(t)-{\mathcal{S}}_{2}^{m}(v)(t)|\leq({\mathcal{L}}^{m}|u-v|)(t)\leq\|({\mathcal{L}}^{m}|u-v|)\|_{\infty}\leq\|{\mathcal{L}}^{m}\|\|u-v\|_{\infty}\leq K_{2}^{m}\gamma_{m}(p_{2})\|u-v\|_{\infty},

(A.24)

for all $t\in[t_{0},T]$ , by Lemma 41. This proves (3.22) with $\varrho_{m}:=K_{2}^{m}\gamma_{m}(p_{2})$ . If there exists $m_{i}\geq 1$ such that $\varrho_{m_{i}}<1$ , one completes the proof of the theorem by the Banach fixed-point theorem [3, Theorem 2.4].

A.4 Proof of Lemma 16

For clarity, we split the proof into several steps. Throughout this section, we set ${\mathcal{X}}:=L^{2}((t_{0},T);{\mathbb{R}}^{k})$ , ${\mathcal{Y}}:=L^{\infty}((t_{0},T);{\mathbb{R}}^{k})$ , and let ${\mathcal{F}}(C_{i})$ be defined by (3.17). Fix $u\in{\mathcal{Y}}$ , and consider the following linear and bounded operators $H_{u,\tau_{i}}$ , $W_{u,\tau_{i}}$ , and $J_{u,\tau_{i}}$ defined by

(H_{u,\tau_{i}}h)(t)=R_{u}(\tau_{i},t)B(t,x_{u}(t))h(t),\qquad(W_{u,\tau_{i}}h)(t)=R_{u}(t,\tau_{i})\int_{t_{0}}^{t}h(s)\,ds,

and

J_{u,\tau_{i}}h=\int_{t_{0}}^{T}D^{2}\Phi_{t,\tau_{i}}(x_{u}(t))B(t,x_{u}(t))u(t)h(t)\,dt+\int_{t_{0}}^{T}D\Phi_{t,\tau_{i}}(x_{u}(t))D_{x}B(t,x_{u}(t))u(t)h(t)\,dt.

Here $R_{u}(t,s)$ is defined in Lemma 8. Since $R_{u}$ is a state transition matrix, one has

(W_{u,\tau_{i}}H_{u,\tau_{i}}h)(t)=R_{u}(t,\tau_{i})\int_{t_{0}}^{t}(H_{u,\tau_{i}}h)(s)\,ds=\int_{t_{0}}^{t}R_{u}(t,s)B(s,x_{u}(s))h(s)\,ds.

We set $K_{u,\tau_{i}}:=J_{u,\tau_{i}}W_{u,\tau_{i}}H_{u,\tau_{i}}$ . It is clear by Assumptions 2 and 5 that ${\mathcal{E}}\in C^{1}({\mathcal{Y}};{\mathbb{R}}^{d})$ ; see, for instance, [4, Theorem 3.2.6]. Hence $G_{\tau_{i}}\in C^{1}({\mathcal{Y}};{\mathbb{R}}^{d})$ by composition. Moreover, for any $h\in{\mathcal{X}}$ ,

D{\mathcal{E}}(u)h=\int_{t_{0}}^{T}R_{u}(T,t)B(t,x_{u}(t))h(t)\,dt,\qquad DG_{\tau_{i}}(u)h=L_{u,\tau_{i}}h+K_{u,\tau_{i}}h.

(A.25)

Using the forward–backward representation (3.2) with $i=2$ , one obtains

{\mathcal{E}}(u)=x_{u}(T)=\Phi_{t_{0},T}(x^{0})+\int_{t_{0}}^{T}D\Phi_{t,T}(x_{u}(t))B(t,x_{u}(t))u(t)\,dt=G_{2}(u)+\Phi_{t_{0},T}(x^{0})+y_{2},

and therefore $D{\mathcal{E}}(u)=DG_{2}(u)$ . The case $i=1$ follows similarly from the same representation, which gives $DG_{\tau_{i}}(u)=D\Phi_{T,\tau_{i}}(x_{u}(T))\,D{\mathcal{E}}(u)$ . This proves (3.26).

Next, suppose that ${\mathcal{N}}_{\tau_{i}}(u)$ is invertible. Then, the canonical right inverse of $L_{u,\tau_{i}}$ satisfies

R_{u,\tau_{i}}:=L_{u,\tau_{i}}^{\ast}{\mathcal{N}}_{\tau_{i}}(u)^{-1}\in\mathscr{L}({\mathbb{R}}^{d},{\mathcal{X}}),\qquad\,L_{u,\tau_{i}}R_{u,\tau_{i}}={\operatorname{Id}}.

Hence, if ${\operatorname{Id}}+A_{u,\tau_{i}}$ is invertible with $A_{u,\tau_{i}}:=K_{u,\tau_{i}}L_{u,\tau_{i}}^{\ast}{\mathcal{N}}_{\tau_{i}}(u)^{-1}$ , one uses $DG_{\tau_{i}}(u)=L_{u,\tau_{i}}+K_{u,\tau_{i}}$ to get

(L_{u,\tau_{i}}+K_{u,\tau_{i}})\,R_{u,\tau_{i}}\,({\operatorname{Id}}+A_{u,\tau_{i}})^{-1}={\operatorname{Id}}\quad\text{which is equivalent to}\quad DG_{\tau_{i}}(u)\,L_{u,\tau_{i}}^{\ast}\begin{bmatrix}DG_{\tau_{i}}(u)\,L_{u,\tau_{i}}^{\ast}\end{bmatrix}^{-1}={\operatorname{Id}}

(A.26)

showing that $DG_{\tau_{i}}(u)\,L_{u,\tau_{i}}^{\ast}\in{\mathbb{R}}^{d\times d}$ is invertible and $DG_{\tau_{i}}(u)$ is right-invertible. So $DG_{\tau_{i}}(u)$ is onto, which implies that $M(u):=DG_{\tau_{i}}(u)DG_{\tau_{i}}(u)^{\ast}$ is invertible.

Suppose now that $M(u):=DG_{\tau_{i}}(u)DG_{\tau_{i}}(u)^{\ast}$ is invertible. The canonical right-inverse of $DG_{\tau_{i}}(u)$ satisfies

T_{u,\tau_{i}}:=DG_{\tau_{i}}(u)^{\ast}M(u)^{-1}\in\mathscr{L}({\mathbb{R}}^{d},{\mathcal{X}}),\qquad\,DG_{\tau_{i}}(u)T_{u,\tau_{i}}={\operatorname{Id}}.

If ${\operatorname{Id}}-A_{u,\tau_{i}}$ is invertible with $A_{u,\tau_{i}}:=K_{u,\tau_{i}}DG_{\tau_{i}}(u)^{\ast}M(u)^{-1}$ , one uses $L_{u,\tau_{i}}=DG_{\tau_{i}}(u)-K_{u,\tau_{i}}$ to obtain

(DG_{\tau_{i}}(u)-K_{u,\tau_{i}})\,T_{u,\tau_{i}}\,({\operatorname{Id}}-A_{u,\tau_{i}})^{-1}={\operatorname{Id}}\quad\Longleftrightarrow\quad L_{u,\tau_{i}}\,DG_{\tau_{i}}(u)^{\ast}\begin{bmatrix}L_{u,\tau_{i}}\,DG_{\tau_{i}}(u)^{\ast}\end{bmatrix}^{-1}={\operatorname{Id}}.

(A.27)

This shows that $L_{u,\tau_{i}}\,DG_{\tau_{i}}(u)^{\ast}\in{\mathbb{R}}^{d\times d}$ is invertible and $L_{u,\tau_{i}}$ is right-invertible. So $L_{u,\tau_{i}}$ is onto, which implies that ${\mathcal{N}}_{\tau_{i}}(u):=L_{u,\tau_{i}}\,L_{u,\tau_{i}}^{\ast}$ is invertible. This completes the proof of the lemma.

A.5 Proof of Theorem 17

In this section, we use the same notations introduced in Section A.4.

Proof of Theorem 17.

Since ${\operatorname{Id}}+K_{u,\tau_{i}}L_{u,\tau_{i}}^{\ast}{\mathcal{N}}_{\tau_{i}}(u)^{-1}$ and ${\mathcal{N}}_{\tau_{i}}(u)$ are invertible for $u\in{\mathcal{F}}(y_{i})\subset{\mathcal{F}}(C_{i})\subset L^{\infty}\subset L^{2}$ , one deduces that $DG_{\tau_{i}}(u)$ is onto by Lemma 16. Then, for any local minimizer $\bar{u}$ on $\mathfrak{F}_{i}:=\{u\in{\mathcal{F}}(y_{i}):\ G_{\tau_{i}}(u)=0\}$ , the Lagrange multiplier theorem in Hilbert spaces [32, Theorem 43.D, p. 290] (applied with $F(u)=\tfrac{1}{2}\|u\|_{2}^{2}$ and the submersion $G_{\tau_{i}}:{\mathcal{F}}(y_{i})\to{\mathbb{R}}^{d}$ , $G_{\tau_{i}}(u)=L_{u,\tau_{i}}u-y_{i}$ ) yields $\lambda\in{\mathbb{R}}^{d}$ such that

\bar{u}=DG_{\tau_{i}}(\bar{u})^{\ast}\lambda,\qquad L_{\bar{u},\tau_{i}}\bar{u}=y_{i}.

(A.28)

Since $DG_{\tau_{i}}(\bar{u})$ is onto, one has that $M(\bar{u})=DG_{\tau_{i}}(\bar{u})DG_{\tau_{i}}(\bar{u})^{\ast}\in{\mathbb{R}}^{d\times d}$ is invertible, and Lemma 16 ensures that $L_{\bar{u},\tau_{i}}\,DG_{\tau_{i}}(\bar{u})^{\ast}\in{\mathbb{R}}^{d\times d}$ is invertible. Next, left-multiplying the first identity in (A.28) by $L_{\bar{u},\tau_{i}}$ and using the second one yields

\lambda=\begin{bmatrix}L_{\bar{u},\tau_{i}}\,DG_{\tau_{i}}(\bar{u})^{\ast}\end{bmatrix}^{-1}y_{i}

(A.29)

Substituting back to (A.28) leads to

\bar{u}=P_{\bar{u},\tau_{i}}y_{i}\qquad\text{where}\quad P_{\bar{u},\tau_{i}}:=DG_{\tau_{i}}(\bar{u})^{\ast}\begin{bmatrix}L_{\bar{u},\tau_{i}}\,DG_{\tau_{i}}(\bar{u})^{\ast}\end{bmatrix}^{-1}.

(A.30)

Recall from (A.27) that

L_{\bar{u},\tau_{i}}\,P_{\bar{u},\tau_{i}}={\operatorname{Id}}

showing that $P_{\bar{u},\tau_{i}}$ is a right-inverse of $L_{\bar{u},\tau_{i}}$ . Next, since the canonical right inverse $R_{\bar{u},\tau_{i}}=L_{\bar{u},\tau_{i}}^{\ast}{\mathcal{N}}_{\tau_{i}}(\bar{u})^{-1}$ of $L_{\bar{u},\tau_{i}}$ satisfies $L_{\bar{u},\tau_{i}}R_{\bar{u},\tau_{i}}={\operatorname{Id}}$ , one deduces that $L_{\bar{u},\tau_{i}}\,(P_{\bar{u},\tau_{i}}-R_{\bar{u},\tau_{i}})=0$ , which implies

(P_{\bar{u},\tau_{i}}-R_{\bar{u},\tau_{i}})y\in\ker L_{\bar{u},\tau_{i}}\qquad\forall y\in{\mathbb{R}}^{d}.

Therefore, (A.30) recast as

\bar{u}=R_{\bar{u},\tau_{i}}y_{i}+z_{i},\qquad z_{i}:=(P_{\bar{u},\tau_{i}}-R_{\bar{u},\tau_{i}})y_{i}\in\ker L_{\bar{u},\tau_{i}}.

(A.31)

Now, let us prove the equivalence in the theorem. Assume that

\langle[L_{\bar{u},\tau_{i}}DG_{\tau_{i}}(\bar{u})^{\ast}]^{-1}y_{i},DG_{\tau_{i}}(\bar{u})\,h\rangle_{{\mathbb{R}}^{d}}=0\qquad\forall\,h\in\ker L_{\bar{u},\tau_{i}}

On one hand, for $h\in\ker\,L_{\bar{u},\tau_{i}}$ , one get from (A.30) that $\langle\bar{u},h\rangle_{L^{2}}=\langle[L_{\bar{u},\tau_{i}}DG_{\tau_{i}}(\bar{u})^{\ast}]^{-1}y_{i},DG_{\tau_{i}}(\bar{u})\,h\rangle_{{\mathbb{R}}^{d}}=0$ , which implies that

\bar{u}\in(\ker\,L_{\bar{u},\tau_{i}})^{\perp}.

(A.32)

On the other hand, by (A.31), (A.32) and $R_{\bar{u},\tau_{i}}y_{i}=L_{\bar{u},\tau_{i}}^{\ast}{\mathcal{N}}_{\tau_{i}}(\bar{u})^{-1}y_{i}\in\operatorname{Ran}(L_{\bar{u},\tau_{i}}^{\ast})=(\ker L_{\bar{u},\tau_{i}})^{\perp}$ , one finds that $z_{i}=\bar{u}-R_{\bar{u},\tau_{i}}y_{i}\in(\ker L_{\bar{u},\tau_{i}})^{\perp}$ . Therefore, $z_{i}\in\ker(L_{\bar{u},\tau_{i}})\cap(\ker L_{\bar{u},\tau_{i}})^{\perp}=\{0\}$ so $z_{i}=0$ , and

\bar{u}=R_{\bar{u},\tau_{i}}y_{i}=L_{\bar{u},\tau_{i}}^{\ast}{\mathcal{N}}_{\tau_{i}}(\bar{u})^{-1}y_{i}={\mathcal{S}}_{i}(\bar{u}).

Conversely, if $\bar{u}=L_{\bar{u},\tau_{i}}^{\ast}{\mathcal{N}}_{\tau_{i}}(\bar{u})^{-1}y_{i}$ , then (A.31) yields $P_{\bar{u},\tau_{i}}y_{i}=R_{\bar{u},\tau_{i}}y_{i}\in\operatorname{Ran}(L_{\bar{u},\tau_{i}}^{\ast})=(\ker L_{\bar{u},\tau_{i}})^{\perp}$ . Let $h\in\ker\,L_{\bar{u},\tau_{i}}$ , then

\langle[L_{\bar{u},\tau_{i}}DG_{\tau_{i}}(\bar{u})^{\ast}]^{-1}y_{i},DG_{\tau_{i}}(\bar{u})\,h\rangle_{{\mathbb{R}}^{d}}=\langle[DG_{\tau_{i}}(\bar{u})^{\ast}L_{\bar{u},\tau_{i}}DG_{\tau_{i}}(\bar{u})^{\ast}]^{-1}y_{i}\,h\rangle_{L^{2}}=\langle P_{\bar{u},\tau_{i}}y_{i},h\rangle_{L^{2}}=0.

This completes the proof of the theorem.

A.6 Proof of Proposition 18

Proof.

For ease in notation, set $L^{p}:=L^{p}((t_{0},T);{\mathbb{R}}^{k})$ for $p\in\{1,2,\infty\}$ . By definition, $\mathfrak{F}_{i}\subset\,L^{\infty}\subset\,L^{2}$ . Let $u\in\,L^{2}$ and $(u_{n})\subset\mathfrak{F}_{i}$ are such that $u_{n}\rightharpoonup u$ in $L^{2}$ . Let us show that $u\in\mathfrak{F}_{i}$ .

Step 1

One has $u\in\,L^{\infty}$ and $\|u\|_{\infty}\leq\,\zeta_{i}$ . In fact, since $(u_{n})\subset\,L^{\infty}$ , by Banach–Alaoglu, there exists a subsequence (not relabeled) and $\tilde{u}\in L^{\infty}$ with $\|\tilde{u}\|_{L^{\infty}}\leq\zeta_{i}$ such that $u_{n}\stackrel{{\scriptstyle\ast}}{{\rightharpoonup}}\tilde{u}$ in $L^{\infty}$ , i.e. $\int_{t_{0}}^{T}\varphi(t)^{\top}\,u_{n}(t)\,dt\to\int_{t_{0}}^{T}\varphi(t)^{\top}\tilde{u}(t)\,dt$ for all $\varphi\in L^{1}$ . On the other hand, $u_{n}\rightharpoonup u$ in $L^{2}$ , so $\int_{t_{0}}^{T}\psi(t)^{\top}u_{n}(t)\,dt\to\int_{t_{0}}^{T}\psi(t)^{\top}u(t)\,dt$ for all $\psi\in L^{2}$ . It holds that

\int_{t_{0}}^{T}\phi(t)^{\top}u(t)\,dt=\lim_{n\to\infty}\int_{t_{0}}^{T}\phi(t)^{\top}u_{n}(t)\,dt=\int_{t_{0}}^{T}\phi(t)^{\top}\tilde{u}(t)\,dt\qquad\forall\phi\in\,L^{2}\cap\,L^{1}=L^{2}.

In particular, for $\phi:=u-\tilde{u}\in\,L^{2}$ , one finds $\int_{t_{0}}^{T}|u(t)-\tilde{u}(t)|^{2}\,dt=0$ , which implies $u=\tilde{u}$ , a.e., and thus $u\in L^{\infty}$ and $\|u\|_{L^{\infty}}=\|\tilde{u}\|_{L^{\infty}}\leq\zeta_{i}$ .

Step 2

One has $G_{\tau_{i}}(u):=L_{u,\tau_{i}}u-y_{i}=0$ . We split the proof of this fact into three steps.

Step 2.1

For $n\in{\mathbb{N}}$ , and $t\in[t_{0},T]$ , set $\delta_{n}(t)=\int_{t_{0}}^{t}B(s,x_{u}(s))\,(u_{n}(s)-u(s))\,ds$ . It is clear that $(\delta_{n})\subset\,C^{0}([t_{0},T];{\mathbb{R}}^{d})$ . Consider the bounded and linear operator $T_{t}:L^{2}\to{\mathbb{R}}^{d}$ , $T_{t}h=\int_{t_{0}}^{T}\mathbf{1}_{[t_{0},t]}(s)\,B(s,x_{u}(s))\,h(s)\,ds$ . Its adjoint $T_{t}^{\ast}:{\mathbb{R}}^{d}\to\,L^{2}$ is $T_{t}^{\ast}\,z=\mathbf{1}_{[t_{0},t]}(\cdot)\,B(\cdot,x_{u}(\cdot))^{\top}\,z\in\,L^{2}$ for all $z\in{\mathbb{R}}^{d}$ . Then $\delta_{n}(t)=T_{t}(u_{n}-u)$ and the weak convergence in $L^{2}$ gives $\langle\delta_{n}(t),z\rangle_{{\mathbb{R}}^{d}}=\langle\,u_{n}-u,T_{t}^{\ast}\,z\rangle_{L^{2}}\to\,0$ for all $z\in{\mathbb{R}}^{d}$ , hence $\delta_{n}(t)\to\,0$ in ${\mathbb{R}}^{d}$ .

For all $n\in{\mathbb{N}}$ , one has by Cauchy-Schwarz inequality,

|\delta_{n}(t)|\leq\|B\|_{\infty}\,(\zeta_{i}+\|u\|_{2})\,\sqrt{\Delta t_{0}}\qquad\forall\,t\in[t_{0},T]

showing that $(\delta_{n})_{n}$ is uniformly bounded. For $(t,t^{\prime})\in[t_{0},T]^{2}$ with $t\geq t^{\prime}$ (the case of $t<t^{\prime}$ is identical), one has by Cauchy-Schwarz inequality,

|\delta_{n}(t)-\delta_{n}(t^{\prime})|\leq\|B\|_{\infty}\,(\zeta_{i}+\|u\|_{2})\,\sqrt{t-t^{\prime}}\qquad\forall\,n\in{\mathbb{N}}

showing that $(\delta_{n})_{n}$ is equicontinuous. Consequently, $(\delta_{n})\subset\,C^{0}([t_{0},T];{\mathbb{R}}^{d})$ is relatively compact, and by Ascoli-Arzèla, it admits a uniformly convergent subsubsequence; its limit must be the pointwise limit, viz.

\|\delta_{n}\|_{\infty}\to\,0\quad\text{as}\quad\,n\to\infty.

Step 2.2

Let $x_{u_{n}}$ and $x_{u}$ be the solution of $\dot{x}=N_{t}(x)+B(t,x)\,u_{n}$ and $\dot{x}=N_{t}(x)+B(t,x)\,u$ , respectively with the same intitial state $x^{0}\in{\mathbb{R}}^{d}$ . Gronwall’s lemma yields

\|x_{u_{n}}-x_{u}\|_{\infty}\leq\Delta t_{0}\,e^{(\Lambda_{1}+L_{B}\zeta_{i})\Delta t_{0}}\|\delta_{n}\|_{\infty}\to\,0\quad\text{as}\quad\,n\to\infty.

Step 2.3

Write $K_{i}(u)(t):=D\Phi_{t,\tau_{i}}(x_{u}(t))\,B(t,x_{u}(t))\in{\mathbb{R}}^{d\times k}$ . Lemma 4 and (A.12) yield

\|K_{i}(u_{n})-K_{i}(u)\|_{\infty}\leq\,\bigg(\frac{\Lambda_{2}}{\Lambda_{1}}\,E_{2,\infty}\|B\|_{\infty}+L_{B}\,E_{1,\infty}\bigg)\|x_{u_{n}}-x_{u}\|_{\infty}\to\,0\quad\text{as}\quad\,n\to\infty.

Overall,

	$\displaystyle G_{\tau_{i}}(u_{n})-G_{\tau_{i}}(u)$	$\displaystyle=\int_{t_{0}}^{T}\!\big(K_{i}(u_{n})-K_{i}(u)\big)\,u_{n}\,dt\;+\;\int_{t_{0}}^{T}\!K_{i}(u)\,(u_{n}-u)\,dt$
		$\displaystyle\xrightarrow[n\to\infty]{}0+0,$

since the first term vanishes by $\|K_{i}(u_{n})-K_{i}(u)\|_{L^{\infty}}\to 0$ and $L^{2}$ -boundedness of $(u_{n})$ , and the second by weak convergence against the fixed $K_{i}(u)\in L^{\infty}\subset\,L^{2}$ . As $G_{\tau_{i}}(u_{n})=0$ , we obtain $G_{\tau_{i}}(u)=0$ .

Step 3

One has $\lambda_{\min}\big({\mathcal{N}}_{\tau_{i}}(u)\big)\geq C_{i}^{-1}$ . In fact, ${\mathcal{N}}_{\tau_{i}}(u)=\int_{t_{0}}^{T}K_{i}(u)(t)\,K_{i}(u)(t)^{\top}\,dt$ , and spectral stability,

\big|\lambda_{\min}({\mathcal{N}}_{\tau_{i}}(u_{n}))-\lambda_{\min}({\mathcal{N}}_{\tau_{i}}(u))\big|\ \leq\|{\mathcal{N}}_{\tau_{i}}(u_{n})-{\mathcal{N}}_{\tau_{i}}(u)\|\leq 2\Delta t_{0}\,\|B\|_{\infty}\,e^{\Lambda_{1}\Delta t_{0}}\,\|K_{i}(u_{n})-K_{i}(u)\|_{\infty}\ \xrightarrow[n\to\infty]{}0.

Since $\lambda_{\min}({\mathcal{N}}_{\tau_{i}}(u_{n}))\geq C_{i}^{-1}$ for all $n$ , we obtain $\lambda_{\min}({\mathcal{N}}_{\tau_{i}}(u))\geq C_{i}^{-1}$ .

If $(u_{n})\subset\mathfrak{F}_{i}$ is a minimizing sequence, then $(\tfrac{1}{2}\|u_{n}\|_{L^{2}}^{2})_{n}$ is a bounded sequence. After extraction, $u_{n_{j}}\rightharpoonup\bar{u}$ in $L^{2}$ . By weak sequential closedness, $\bar{u}\in\mathfrak{F}_{i}$ . Then $\tfrac{1}{2}\|\bar{u}\|_{L^{2}}^{2}\leq\liminf_{j}\tfrac{1}{2}\|u_{n_{j}}\|_{L^{2}}^{2}$ , so $\bar{u}$ attains the infimum.

A.7 Proof of Proposition 23

Proof.

Let $(x^{0},\,u)\in{\mathbb{R}}^{d}\times L^{\infty}((t_{0},T);{\mathbb{R}}^{k})$ and $x_{u}$ be the solution to ( Σ ). Then, applying (2.7) with $\beta(t)=\Phi_{t,t_{0}}(x_{u}(t))$ , one immediately finds that

|D\Phi_{t,\tau_{i}}(x_{u}(t))z|\geq e^{-\Lambda_{1}|\tau_{i}-t|}|z|\qquad\forall(t,\,z)\in[t_{0},T]\times{\mathbb{R}}^{d},\quad\tau_{1}=t_{0},\,\tau_{2}=T.

It follows that

y^{\top}{\mathcal{N}}_{\tau_{i}}(u)\,y=\int_{t_{0}}^{T}|D\Phi_{t,\tau_{i}}(x_{u}(t))^{\top}\,B(t,x_{u}(t))^{\top}y|^{2}\,dt\geq e^{-2\Lambda_{1}\Delta t_{0}}\,|y|^{2}\int_{t_{0}}^{T}b(t)^{2}\,dt\geq\frac{e^{-2\Lambda_{1}\Delta t_{0}}\,\|b\|_{1}^{2}}{\Delta t_{0}}\,|y|^{2},

by the Cauchy-Schwarz inequality. This proves (3.32). So, letting $C_{i}>0$ as in the proposition, one gets

{\mathcal{F}}(C_{i})=\left\{u\in L^{\infty}((t_{0},T);{\mathbb{R}}^{k}):\lambda_{\min}\big({\mathcal{N}}_{\tau_{i}}(u)\big)\geq\,e^{-2\Lambda_{1}\Delta t_{0}}\|b\|_{1}^{2}/\Delta t_{0}\right\}=L^{\infty}((t_{0},T);{\mathbb{R}}^{k}),

and (3.21) is automatically satisfied.

A.8 Proof of Theorem 24

Proof.

Let $C_{i}>0$ as in the statement of the proposition, and $y_{i}\in{\mathbb{R}}^{d}$ . Then, (3.21) is satisfied if and only if $v\in{\mathcal{F}}(C_{i})$ for every $v\in{\mathcal{S}}_{i}({\mathcal{F}}(y_{i}))$ . Observe that

{\mathcal{F}}(C_{i})=\left\{u\in L^{\infty}((t_{0},T);{\mathbb{R}}^{k}):\lambda_{\min}\big({\mathcal{N}}_{\tau_{i}}(u)\big)\geq\lambda_{\min}\big({\mathcal{N}}_{\tau_{i}}(u_{i}^{\flat})\big)/(1+\theta)\right\}.

Now, $v\in{\mathcal{S}}_{i}({\mathcal{F}}(y_{i}))$ if and only if $v={\mathcal{S}}_{i}(u)=B(t,x_{u}(t))^{\top}D\Phi_{t,\tau_{i}}(x_{u}(t))^{\top}\,{\mathcal{N}}_{\tau_{i}}(u)^{-1}\,y_{i}$ for some $u\in{\mathcal{F}}(C_{i})$ , where $\tau_{1}:=t_{0}$ and $\tau_{2}:=T$ . Therefore, $\|v\|_{\infty}\leq\|B\|_{\infty}\,e^{\Lambda_{1}\Delta t_{0}}\lambda_{\min}^{-1}({\mathcal{N}}_{\tau_{i}}(u))\,|y_{i}|\leq(1+\theta)\,e^{\Lambda_{1}\Delta t_{0}}\|B\|_{\infty}|y_{i}|/\lambda_{\min}\big({\mathcal{N}}_{\tau_{i}}(u_{i}^{\flat})\big)$ . It remains to prove that

\lambda_{\min}({\mathcal{N}}_{\tau_{i}}(v))=\lambda_{\min}\big({\mathcal{N}}_{\tau_{i}}({\mathcal{S}}_{i}(u))\big)\geq\lambda_{\min}\big({\mathcal{N}}_{\tau_{i}}(u_{i}^{\flat})\big)/(1+\theta).

(A.33)

By the spectral stability and $L_{{\mathcal{N}}_{\tau_{i}}}$ -Lipschitz of $u\in\{u\in\,L^{\infty}((t_{0},T);{\mathbb{R}}^{k}):\|u\|\leq\,\zeta_{i}\}\mapsto{\mathcal{N}}_{\tau_{i}}(u)$ , one finds that

\left|\lambda_{\min}\big({\mathcal{N}}_{\tau_{i}}({\mathcal{S}}_{i}(u))\big)-\lambda_{\min}\big({\mathcal{N}}_{\tau_{i}}(u_{i}^{\flat})\big)\right|\leq L_{{\mathcal{N}}_{\tau_{i}}}\left(\frac{(1+\theta)\,e^{\Lambda_{1}\Delta t_{0}}\|B\|_{\infty}|y_{i}|}{\lambda_{\min}\big({\mathcal{N}}_{\tau_{i}}(u_{i}^{\flat})\big)}+\|u_{i}^{\flat}\|_{\infty}\right).

Since

\left|\lambda_{\min}\big({\mathcal{N}}_{\tau_{i}}({\mathcal{S}}_{i}(u))\big)-\lambda_{\min}\big({\mathcal{N}}_{\tau_{i}}(u_{i}^{\flat})\big)\right|\geq\frac{\lambda_{\min}\big({\mathcal{N}}_{\tau_{i}}(u_{i}^{\flat})\big)}{1+\theta}+\frac{\theta\lambda_{\min}\big({\mathcal{N}}_{\tau_{i}}(u_{i}^{\flat})\big)}{1+\theta}-\lambda_{\min}\big({\mathcal{N}}_{\tau_{i}}({\mathcal{S}}_{i}(u))\big),

one deduces that if

\frac{\theta\lambda_{\min}\big({\mathcal{N}}_{\tau_{i}}(u_{i}^{\flat})\big)}{1+\theta}\geq L_{{\mathcal{N}}_{\tau_{i}}}\left(\frac{(1+\theta)\,e^{\Lambda_{1}\Delta t_{0}}\|B\|_{\infty}|y_{i}|}{\lambda_{\min}\big({\mathcal{N}}_{\tau_{i}}(u_{i}^{\flat})\big)}+\|u_{i}^{\flat}\|_{\infty}\right)

(A.34)

then (A.33) is satisfied. Since (A.34) is equivalent to (3.34), this completes the proof of the proposition.

A.9 Proof of Lemma 25

Proof.

Write $K_{i}(u)(t):=D\Phi_{t,\tau_{i}}(x_{u}(t))\,B(t,x_{u}(t))\in{\mathbb{R}}^{d\times k}$ so that ${\mathcal{N}}_{\tau_{i}}(u)=\int_{t_{0}}^{T}K_{i}(u)(t)\,K_{i}(u)(t)^{\top}\,dt$ . Clearly, ${\mathcal{N}}_{\tau_{i}}$ is continuous on $L^{\infty}$ by Lebesgue dominated convergence theorem. Let us prove the second part of the lemma. Let $u,\,v\in\{u\in\,L^{\infty}((t_{0},T);{\mathbb{R}}^{k}):\|u\|_{\infty}\leq\,\zeta_{i}\}$ . By the Cauchy-Schwarz inequality, one finds that

\frac{d}{dt}|x_{u}(t)-x_{v}(t)|=|\dot{x}_{u}(t)-\dot{x}_{v}(t)|\leq(\Lambda_{1}+L_{B}\,\zeta_{i})|x_{u}(t)-x_{v}(t)|+\|B\|_{\infty}\,|u(t)-v(t)|

which, by Gronwall’s lemma, yields

|x_{u}(t)-x_{v}(t)|\leq\|B\|_{\infty}\frac{e^{(\Lambda_{1}+L_{B}\,\zeta_{i})(t-t_{0})}-1}{(\Lambda_{1}+L_{B}\,\zeta_{i})}\|u-v\|_{\infty}\qquad\forall\in[t_{0},T].

(A.35)

Using Lemma 4, one finds (we let $\Delta t_{0}:=T-t_{0}$ )

$\displaystyle\frac{\\|{\mathcal{N}}_{\tau_{2}}(u)-{\mathcal{N}}_{\tau_{2}}(v)\\|}{\\|u-v\\|_{\infty}}$	$\displaystyle\leq$	$\displaystyle\frac{2\\|B\\|_{\infty}^{3}\Lambda_{2}}{\Lambda_{1}(\Lambda_{1}+L_{B}\,\zeta_{2})}\int_{t_{0}}^{T}\,e^{\Lambda_{1}(T-t)}\,\left(e^{2\Lambda_{1}(T-t)}-e^{\Lambda_{1}(T-t)}\right)\,(e^{(\Lambda_{1}+L_{B}\,\zeta_{2})(t-t_{0})}-1)\,dt$
		$\displaystyle+\frac{2L_{B}\\|B\\|_{\infty}^{2}}{(\Lambda_{1}+L_{B}\,\zeta_{2})}\int_{t_{0}}^{T}e^{2\Lambda_{1}(T-t)}(e^{(\Lambda_{1}+L_{B}\,\zeta_{2})(t-t_{0})}-1)\,dt$
	$\displaystyle=$	$\displaystyle\frac{L_{B}\\|B\\|_{\infty}^{2}e^{\Lambda_{1}\Delta t_{0}}}{\Lambda_{1}}\left[\frac{e^{\Lambda_{1}\Delta t_{0}}-e^{L_{B}\zeta_{2}\Delta t_{0}}}{\Lambda_{1}-L_{B}\zeta_{i}}-\frac{e^{L_{B}\zeta_{2}\Delta t_{0}}-e^{-\Lambda_{1}\Delta t_{0}}}{L_{B}\zeta_{2}+\Lambda_{1}}\right]+$
		$\displaystyle\hskip-56.9055pt\frac{2\\|B\\|_{\infty}^{3}\Lambda_{2}e^{\Lambda_{1}\Delta t_{0}}}{\Lambda_{1}(\Lambda_{1}+L_{B}\,\zeta_{2})}\left[\frac{e^{2\Lambda_{1}\Delta t_{0}}-e^{L_{B}\zeta_{2}\Delta t_{0}}}{2\Lambda_{1}-L_{B}\zeta_{2}}+\frac{e^{\Lambda_{1}\Delta t_{0}}-e^{L_{B}\zeta_{2}\Delta t_{0}}}{\Lambda_{1}-L_{B}\zeta_{2}}+\frac{e^{\Lambda_{1}\Delta t_{0}}-e^{-\Lambda_{1}\Delta t_{0}}}{2\Lambda_{1}}-\frac{e^{2\Lambda_{1}\Delta t_{0}}-e^{-\Lambda_{1}\Delta t_{0}}}{3\Lambda_{1}}\right],$

by a direct integration. The same proof applies to ${\mathcal{N}}_{\tau_{1}}$ .

A.10 Proof of Proposition 33

Proof.

Recall that $DN(x)=-D+W\,D\sigma(x)$ and $D^{2}N(x)=W\,D^{2}\sigma(x)$ . Then, from

\frac{d}{dt}\phi_{t}(x)=N(\phi_{t}(x)),\quad\phi_{0}(x)=x;\qquad\frac{d}{dt}D\phi_{t}(x)=DN(\phi_{t}(x))\,D\phi_{t}(x),\quad D\phi_{0}(x)={\operatorname{Id}}\quad\forall x\in{\mathbb{R}}^{d},

Cauchy-Schwarz inequality yields

\frac{d}{dt}|D\phi_{t}(x)\,y|=\frac{\langle\,-D\,D\phi_{t}(x)\,y,D\phi_{t}(x)\,y\rangle+\langle\,W\,D\sigma(\phi_{t}(x))\,D\phi_{t}(x)\,y,D\phi_{t}(x)\,y\rangle}{|D\phi_{t}(x)\,y|}\leq\Gamma\,|D\phi_{t}(x)\,y|\quad\forall y\in{\mathbb{R}}^{d}\char 92\relax\{0\},

which, by Gronwall’s lemma, leads to

\|D\phi_{t}(x)\|\leq e^{\Gamma t}\qquad\forall\,x\in{\mathbb{R}}^{d},\,\forall t\geq 0.

(A.36)

Similarly, one has

\frac{d}{dt}D^{2}\phi_{t}(x)=D^{2}N(\phi_{t}(x))\,\left[D\phi_{t}(x),D\phi_{t}(x)\right]+DN(\phi_{t}(x))\,D^{2}\phi_{t}(x),\quad D^{2}\phi_{0}(x)=0,\quad x\in{\mathbb{R}}^{d},

so that letting $g(t):=D^{2}\phi_{t}(x)[y,z]$ for $y,z\in{\mathbb{R}}^{d}$ , Cauchy-Schwarz inequality and (A.36) lead to

	$\displaystyle\frac{d}{dt}\|g(t)\|$	$\displaystyle=$	$\displaystyle\frac{\langle\,-D\,g(t),g(t)\rangle+\langle\,W\,D\sigma(\phi_{t}(x))\,g(t),g(t)\rangle+\langle\,W\,D^{2}\sigma(\phi_{t}(x))\left[D\phi_{t}(x)y,D\phi_{t}(x)z\right],g(t)\rangle}{\|g(t)\|}$
		$\displaystyle\leq$	$\displaystyle\Gamma\,\|g(t)\|+\Gamma_{2}\,e^{2\Gamma\,t}\,\|y\|\,\|z\|,$

which, by Gronwall’s lemma and immediate integration, yields

\|D^{2}\phi_{t}(x)\|\leq\begin{cases}\Gamma_{2}\,\frac{e^{2\Gamma\,t}-e^{\Gamma\,t}}{\Gamma}&\quad\text{if}\quad\Gamma\neq 0,\cr\Gamma_{2}\,t&\quad\text{if}\quad\Gamma=0,\end{cases}\qquad\forall\,x\in{\mathbb{R}}^{d},\,\forall t\geq 0.

(A.37)

In particular as in Lemma 4, one can replace $x\in{\mathbb{R}}^{d}$ in (A.36) and (A.37) with $\beta(\cdot)$ for any $\beta\in C^{0}([0,T];{\mathbb{R}}^{d})$ . Finally, for $u,v\in L^{\infty}((0,T);{\mathbb{R}}^{k})$ , one uses the same technique to prove (A.36) and (A.37), and get

|x_{u}(t)-x_{v}(t)|\leq\begin{cases}\|B\|_{\infty}\,\frac{e^{\Gamma\,t}-1}{\Gamma}\|u-v\|_{\infty}&\quad\text{if}\quad\Gamma\neq 0,\cr\|B\|_{\infty}\,t\,\|u-v\|_{\infty}&\quad\text{if}\quad\Gamma=0,\end{cases}\qquad\forall t\geq 0.

(A.38)

Now, recall that ${\mathcal{N}}_{\tau_{2}}(u)=\int_{0}^{T}\,D\phi_{t}(x_{u}(T-t))\,B(T-t)\,B(T-t)^{\top}\,D\phi_{t}(x_{u}(T-t))^{\top}\,dt$ , so that letting $Q_{u}(t):=D\phi_{t}(x_{u}(T-t))$ , if $\Gamma=0$ , the estimates (A.36), (A.37) and (A.38) yield

\|{\mathcal{N}}_{\tau_{2}}(u)-{\mathcal{N}}_{\tau_{2}}(v)\|\leq\,2\,\|B\|_{\infty}^{3}\,\Gamma_{2}\|u-v\|_{\infty}\,\int_{0}^{T}\,t\,(T-t)\,dt=\frac{\Gamma_{2}\,\|B\|_{\infty}^{3}\,T^{3}}{3}\,\|u-v\|_{\infty}.

(A.39)

When $\Gamma\neq 0$ , one finds that

\|{\mathcal{N}}_{\tau_{2}}(u)-{\mathcal{N}}_{\tau_{2}}(v)\|\leq\,2\,\|B\|_{\infty}^{3}\,\Gamma_{2}\,\|u-v\|_{\infty}\,\int_{0}^{T}\,\frac{(e^{3\Gamma\,t}-e^{2\Gamma\,t})(e^{\Gamma(T-t)}-1)}{\Gamma^{2}}\,dt=\frac{\Gamma_{2}\,\|B\|_{\infty}^{3}}{3}\left(\frac{e^{\Gamma\,T}-1}{\Gamma}\right)^{3}\,\|u-v\|_{\infty},

which completes the proof of the inequality involving ${\mathcal{N}}_{\tau_{2}}$ . The same arguments are used to prove those of ${\mathcal{N}}_{\tau_{1}}$ .

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$\displaystyle\frac{\\|{\mathcal{N}}_{\tau_{1}}(u)-{\mathcal{N}}_{\tau_{1}}(v)\\|}{\\|u-v\\|_{\infty}}$	$\displaystyle\leq\,L_{{\mathcal{N}}_{\tau_{1}}}:=\frac{L_{B}\\|B\\|_{\infty}^{2}e^{\Lambda_{1}\Delta t_{0}}}{\Lambda_{1}}\Bigg[\frac{e^{(2\Lambda_{1}+L_{B}\zeta_{1})\Delta t_{0}}-e^{\Lambda_{1}\Delta t_{0}}}{\Lambda_{1}+L_{B}\zeta_{1}}+\frac{e^{(2\Lambda_{1}+L_{B}\zeta_{1})\Delta t_{0}}-e^{-\Lambda_{1}\Delta t_{0}}}{L_{B}\zeta_{1}+3\Lambda_{1}}\Bigg]$
	$\displaystyle\quad\qquad+\frac{2\\|B\\|_{\infty}^{3}\Lambda_{2}e^{\Lambda_{1}\Delta t_{0}}}{\Lambda_{1}(\Lambda_{1}+L_{B}\zeta_{1})}\Bigg[\frac{e^{(3\Lambda_{1}+L_{B}\zeta_{1})\Delta t_{0}}-e^{-\Lambda_{1}\Delta t_{0}}}{4\Lambda_{1}+L_{B}\zeta_{1}}-\frac{e^{(2\Lambda_{1}+L_{B}\zeta_{1})\Delta t_{0}}-e^{-\Lambda_{1}\Delta t_{0}}}{3\Lambda_{1}+L_{B}\zeta_{1}}$
	$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+\frac{e^{\Lambda_{1}\Delta t_{0}}-e^{-\Lambda_{1}\Delta t_{0}}}{2\Lambda_{1}}-\frac{e^{2\Lambda_{1}\Delta t_{0}}-e^{-\Lambda_{1}\Delta t_{0}}}{3\Lambda_{1}}\Bigg].$	(3.35)

$\displaystyle\frac{\\|{\mathcal{N}}_{\tau_{2}}(u)-{\mathcal{N}}_{\tau_{2}}(v)\\|}{\\|u-v\\|_{\infty}}$	$\displaystyle\leq\,L_{{\mathcal{N}}_{\tau_{2}}}:=\frac{L_{B}\\|B\\|_{\infty}^{2}e^{\Lambda_{1}\Delta t_{0}}}{\Lambda_{1}}\Bigg[\frac{e^{\Lambda_{1}\Delta t_{0}}-e^{L_{B}\zeta_{2}\Delta t_{0}}}{\Lambda_{1}-L_{B}\zeta_{2}}-\frac{e^{L_{B}\zeta_{2}\Delta t_{0}}-e^{-\Lambda_{1}\Delta t_{0}}}{L_{B}\zeta_{2}+\Lambda_{1}}\Bigg]$
	$\displaystyle\quad\qquad\qquad+\frac{2\\|B\\|_{\infty}^{3}\Lambda_{2}e^{\Lambda_{1}\Delta t_{0}}}{\Lambda_{1}(\Lambda_{1}+L_{B}\zeta_{2})}\Bigg[\frac{e^{2\Lambda_{1}\Delta t_{0}}-e^{L_{B}\zeta_{2}\Delta t_{0}}}{2\Lambda_{1}-L_{B}\zeta_{2}}+\frac{e^{\Lambda_{1}\Delta t_{0}}-e^{L_{B}\zeta_{2}\Delta t_{0}}}{\Lambda_{1}-L_{B}\zeta_{2}}$
	$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\frac{e^{\Lambda_{1}\Delta t_{0}}-e^{-\Lambda_{1}\Delta t_{0}}}{2\Lambda_{1}}-\frac{e^{2\Lambda_{1}\Delta t_{0}}-e^{-\Lambda_{1}\Delta t_{0}}}{3\Lambda_{1}}\Bigg].$	(3.36)

$\displaystyle\frac{\\|{\mathcal{N}}_{\tau_{2}}(u)-{\mathcal{N}}_{\tau_{2}}(v)\\|}{\\|u-v\\|_{\infty}}$	$\displaystyle\leq$	$\displaystyle\frac{2\\|B\\|_{\infty}^{3}\Lambda_{2}}{\Lambda_{1}(\Lambda_{1}+L_{B}\,\zeta_{2})}\int_{t_{0}}^{T}\,e^{\Lambda_{1}(T-t)}\,\left(e^{2\Lambda_{1}(T-t)}-e^{\Lambda_{1}(T-t)}\right)\,(e^{(\Lambda_{1}+L_{B}\,\zeta_{2})(t-t_{0})}-1)\,dt$
		$\displaystyle+\frac{2L_{B}\\|B\\|_{\infty}^{2}}{(\Lambda_{1}+L_{B}\,\zeta_{2})}\int_{t_{0}}^{T}e^{2\Lambda_{1}(T-t)}(e^{(\Lambda_{1}+L_{B}\,\zeta_{2})(t-t_{0})}-1)\,dt$
	$\displaystyle=$	$\displaystyle\frac{L_{B}\\|B\\|_{\infty}^{2}e^{\Lambda_{1}\Delta t_{0}}}{\Lambda_{1}}\left[\frac{e^{\Lambda_{1}\Delta t_{0}}-e^{L_{B}\zeta_{2}\Delta t_{0}}}{\Lambda_{1}-L_{B}\zeta_{i}}-\frac{e^{L_{B}\zeta_{2}\Delta t_{0}}-e^{-\Lambda_{1}\Delta t_{0}}}{L_{B}\zeta_{2}+\Lambda_{1}}\right]+$
		$\displaystyle\hskip-56.9055pt\frac{2\\|B\\|_{\infty}^{3}\Lambda_{2}e^{\Lambda_{1}\Delta t_{0}}}{\Lambda_{1}(\Lambda_{1}+L_{B}\,\zeta_{2})}\left[\frac{e^{2\Lambda_{1}\Delta t_{0}}-e^{L_{B}\zeta_{2}\Delta t_{0}}}{2\Lambda_{1}-L_{B}\zeta_{2}}+\frac{e^{\Lambda_{1}\Delta t_{0}}-e^{L_{B}\zeta_{2}\Delta t_{0}}}{\Lambda_{1}-L_{B}\zeta_{2}}+\frac{e^{\Lambda_{1}\Delta t_{0}}-e^{-\Lambda_{1}\Delta t_{0}}}{2\Lambda_{1}}-\frac{e^{2\Lambda_{1}\Delta t_{0}}-e^{-\Lambda_{1}\Delta t_{0}}}{3\Lambda_{1}}\right],$

	$\displaystyle\frac{d}{dt}\|g(t)\|$	$\displaystyle=$	$\displaystyle\frac{\langle\,-D\,g(t),g(t)\rangle+\langle\,W\,D\sigma(\phi_{t}(x))\,g(t),g(t)\rangle+\langle\,W\,D^{2}\sigma(\phi_{t}(x))\left[D\phi_{t}(x)y,D\phi_{t}(x)z\right],g(t)\rangle}{\|g(t)\|}$
		$\displaystyle\leq$	$\displaystyle\Gamma\,\|g(t)\|+\Gamma_{2}\,e^{2\Gamma\,t}\,\|y\|\,\|z\|,$

Control analysis and synthesis for general control-affine systemsThanks:

Abstract

keywords

Funding.

MSC

1 Introduction

Notation 1.

Assumption 2.

Remark 3.

2 Prerequisites and flow representation

Lemma 4.

3 Control–affine dynamics with state–dependent and time-varying input matrix

Assumption 5.

3.1 Representation of solutions

Theorem 6.

Corollary 7.

Proof.

Lemma 8.

Remark 9.

3.2 Controllability results

3.2.1 Synthesis, analysis, and optimal control results

Definition 10.

Remark 11.

Proposition 12.

Feasible coercivity class

Remark 13.

Theorem 14.

Remark 15.

Lemma 16.

Theorem 17.

Proposition 18.

Theorem 19.

Proof.

Remark 20 (A posteriori check of the minimizer).

Remark 21.

Remark 22 (Relation to reachability for fixed T>t0T>t_{0}).

Proposition 23.

Theorem 24.

Lemma 25.

Remark 26.

Corollary 27.

Remark 28.

3.2.2 A practical control reference: the zero control

Corollary 29.

Proof.

Remark 30.

3.2.3 Optimizing the admissible radius: balancing coercivity and control size

Proposition 31.

Proof.

Remark 32.

3.2.4 Refined estimates for Hopfield-type dynamics

Proposition 33.

Remark 34.

Example 35.

4 Drift–modulated control-affine dynamics with time-varying and state–dependent input matrix

Assumption 36.

Theorem 37.

Remark 38.

Proof of Theorem 37.

Remark 39.

5 Conclusion

Appendix A Proofs of results stated in the main text

A.1 Proof of Theorem 6

Proof.

A.2 Proof of Lemma 8

Proof.

A.3 Proof of Theorem 14

Lemma 40.

Proof.

Lemma 41.

Proof.

Lemma 42.

Proof.

Proof of Theorem 14.

A.4 Proof of Lemma 16

A.5 Proof of Theorem 17

Proof of Theorem 17.

A.6 Proof of Proposition 18

Proof.

Step 1

Remark 22 (Relation to reachability for fixed $T>t_{0}$ ).