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Coercive ISS-Lyapunov functionals for regular infinite-dimensional systems and applications

Swann Marx

TL;DR

This work tackles the lack of coercivity in ISS-Lyapunov functionals for regular infinite-dimensional systems with outputs. It shows that if the output operator yields exact observability, a noncoercive ISS Lyapunov functional can be augmented by the observability Gramian to become coercive, producing a Lyapunov inequality that includes an explicit output term. The approach relies on regular linear systems theory and constructs the coercive functional via a Gramian-based additive term, enabling applications to singular perturbation analyses and output regulation, with the KdV equation serving as a nontrivial demonstration. The results unify ISS methodology with exactly observable outputs and offer pathways to strictification through observer designs, potentially impacting controller and observer synthesis for distributed-parameter systems.

Abstract

This paper proposes the construction of a coercive ISS-Lyapunov functional for linear regular infinite-dimensional system. Indeed, as already known, Lyapunov functionals for infinite-dimensional systems might be not coercive. Under the assumption that there exists an exactly observable output, we are able to make coercive a Lyapunov functional which is not coercive under additional regularity assumption. We discuss also about the potential applications of such a Lyapunov functional in singular perturbation theory and output regulation. The results are illustrated on a non-trivial equation, namely, the Korteweg-de Vries equation.

Coercive ISS-Lyapunov functionals for regular infinite-dimensional systems and applications

TL;DR

This work tackles the lack of coercivity in ISS-Lyapunov functionals for regular infinite-dimensional systems with outputs. It shows that if the output operator yields exact observability, a noncoercive ISS Lyapunov functional can be augmented by the observability Gramian to become coercive, producing a Lyapunov inequality that includes an explicit output term. The approach relies on regular linear systems theory and constructs the coercive functional via a Gramian-based additive term, enabling applications to singular perturbation analyses and output regulation, with the KdV equation serving as a nontrivial demonstration. The results unify ISS methodology with exactly observable outputs and offer pathways to strictification through observer designs, potentially impacting controller and observer synthesis for distributed-parameter systems.

Abstract

This paper proposes the construction of a coercive ISS-Lyapunov functional for linear regular infinite-dimensional system. Indeed, as already known, Lyapunov functionals for infinite-dimensional systems might be not coercive. Under the assumption that there exists an exactly observable output, we are able to make coercive a Lyapunov functional which is not coercive under additional regularity assumption. We discuss also about the potential applications of such a Lyapunov functional in singular perturbation theory and output regulation. The results are illustrated on a non-trivial equation, namely, the Korteweg-de Vries equation.
Paper Structure (3 sections, 2 theorems, 17 equations)

This paper contains 3 sections, 2 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Construction of a coercive Lyapunov functional
  3. Regular linear systems

Key Result

proposition thmcounterproposition

If $u\in C^2([0,\infty);U)$ and $\in D(S)$, then the system eq:ISS admits a unique strong solution satisfying $z(0) = z_0$.

Theorems & Definitions (5)

  • definition thmcounterdefinition: Well-posed LTI systems
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • proposition thmcounterproposition
  • proposition thmcounterproposition