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Lyapunov characterization of boundedness of reachability sets for infinite-dimensional systems

Patrick Bachmann, Andrii Mironchenko

Abstract

We prove a converse Lyapunov theorem for boundedness of reachability sets for a general class of control systems whose flow is Lipschitz continuous on compact intervals with respect to trajectory-dominated inputs. We show that this condition is satisfied by many semi-linear evolution equations. For ordinary differential equations, as a consequence of our results, we obtain a converse Lyapunov theorem for forward completeness, without a priori restrictions on the magnitude of inputs.

Lyapunov characterization of boundedness of reachability sets for infinite-dimensional systems

Abstract

We prove a converse Lyapunov theorem for boundedness of reachability sets for a general class of control systems whose flow is Lipschitz continuous on compact intervals with respect to trajectory-dominated inputs. We show that this condition is satisfied by many semi-linear evolution equations. For ordinary differential equations, as a consequence of our results, we obtain a converse Lyapunov theorem for forward completeness, without a priori restrictions on the magnitude of inputs.
Paper Structure (12 sections, 12 theorems, 93 equations, 1 figure)

This paper contains 12 sections, 12 theorems, 93 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Abstract control systems
  4. Regularity of solutions
  5. Reachability and behavior for trajectory-dominated inputs
  6. Lyapunov criterion
  7. Semi-linear evolution equations
  8. Sufficient conditions for Lipschitz continuity on compact intervals with respect to trajectory-dominated inputs
  9. Relation of notions of Lipschitz continuity of the flow
  10. Lyapunov characterization of forward completeness for ODE systems
  11. Undisturbed systems
  12. Conclusion

Key Result

Proposition III.1

Let $\Sigma$ be a forward complete control system. The following are equivalent:

Figures (1)

  • Figure 1: Outline of the proof of Theorem \ref{['thm:BRSLyapunovFunction']}.

Theorems & Definitions (39)

  • Definition II.1
  • Remark II.2
  • Definition II.3
  • Definition II.4: MiW18b
  • Definition II.5
  • Definition II.6
  • Definition II.7
  • Proposition III.1: BRS Characterization
  • proof
  • Proposition III.2
  • ...and 29 more