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Control analysis and synthesis for general control-affine systems

Cyprien Tamekue, ShiNung Ching

TL;DR

This work develops a constructive framework for controllability and synthesis of nonlinear control-affine systems by introducing trajectory-dependent Gramian maps ${\mathcal{N}}_i(u)$ that extend the linear Gramian concept. By casting steering as a fixed-point problem on a feasible coercivity class and leveraging Caccioppoli's fixed-point theorem, the authors obtain unique fixed points that provide explicit control laws and energy certificates; under a range-alignment condition these fixed points correspond to global minimal-energy controls on the admissible set. The framework yields verifiable, numerically implementable controllers and supplies structural criteria (e.g., uniform input coercivity, reference-control-based coercivity) to guarantee nonemptiness of the admissible class and actionable reachability results, including refined estimates for Hopfield-type networks. An extension to drift-modulated dynamics via a freezing technique broadens applicability to general nonlinear control-affine systems, with Schaefer’s fixed-point theorem used to handle the general case. The results offer a practical alternative to homotopy methods for nonlinear controllability, delivering inner approximations of reachable sets, energy bounds, and a clear path to windowed, patching-based implementations for broader state-space coverage.

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, synthesis iterates exhibit factorial decay, and the Caccioppoli fixed-point theorem gives a unique fixed point that steers the system and satisfies an energy identity. When, in addition, an annihilation (range-alignment) 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 admissible 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.

Control analysis and synthesis for general control-affine systems

TL;DR

This work develops a constructive framework for controllability and synthesis of nonlinear control-affine systems by introducing trajectory-dependent Gramian maps that extend the linear Gramian concept. By casting steering as a fixed-point problem on a feasible coercivity class and leveraging Caccioppoli's fixed-point theorem, the authors obtain unique fixed points that provide explicit control laws and energy certificates; under a range-alignment condition these fixed points correspond to global minimal-energy controls on the admissible set. The framework yields verifiable, numerically implementable controllers and supplies structural criteria (e.g., uniform input coercivity, reference-control-based coercivity) to guarantee nonemptiness of the admissible class and actionable reachability results, including refined estimates for Hopfield-type networks. An extension to drift-modulated dynamics via a freezing technique broadens applicability to general nonlinear control-affine systems, with Schaefer’s fixed-point theorem used to handle the general case. The results offer a practical alternative to homotopy methods for nonlinear controllability, delivering inner approximations of reachable sets, energy bounds, and a clear path to windowed, patching-based implementations for broader state-space coverage.

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, synthesis iterates exhibit factorial decay, and the Caccioppoli fixed-point theorem gives a unique fixed point that steers the system and satisfies an energy identity. When, in addition, an annihilation (range-alignment) 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 admissible 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.

Paper Structure

This paper contains 29 sections, 25 theorems, 178 equations.

Table of Contents

  1. Introduction
  2. Prerequisites and flow representation
  3. Control–affine dynamics with state–dependent and time-varying input matrix
  4. Representation of solutions
  5. Controllability results
  6. Synthesis, analysis, and optimal control results
  7. Feasible coercivity class
  8. A practical control reference: the zero control
  9. Optimizing the admissible radius: balancing coercivity and control size
  10. Refined estimates for Hopfield-type dynamics
  11. Drift–modulated control-affine dynamics with time-varying and state–dependent input matrix
  12. Conclusion
  13. Proofs of results stated in the main text
  14. Proof of Theorem \ref{['thm:sol representation nonautonomous']}
  15. Proof of Lemma \ref{['lem:state-transition matrix of the nonlinear control B=0']}
  16. ...and 14 more sections

Key Result

Lemma 2.1

Let $\beta\in C^0([t_0, T];{\mathbb R}^d)$ and set $q:=\Lambda_2/\Lambda_1$. It holds for every $(s, t)\in[t_0, T]^2$, $s\le t$, where $D^2\Phi_{s,t}$ is the second derivative (third-order tensor) of $\Phi_{s,t}$.

Theorems & Definitions (64)

  • Remark 1.3
  • Lemma 2.1
  • Theorem 3.2
  • Corollary 3.3
  • Proof 1
  • Lemma 3.4
  • Remark 3.5
  • Definition 3.6
  • Definition 3.7
  • Remark 3.8
  • ...and 54 more