Noncoercive Lyapunov functions for input-to-state stability of infinite-dimensional systems

TL;DR

This paper investigates noncoercive Lyapunov functions for infinite-dimensional systems with inputs, establishing their role in input-to-state stability and providing explicit constructions for certain linear systems, including heat equations.

Contribution

It introduces the significance of noncoercive Lyapunov functions in input-to-state stability analysis for infinite-dimensional systems and offers explicit constructions for specific linear cases.

Findings

Noncoercive Lyapunov functions imply norm-to-integral input-to-state stability.

Under mild regularity, input-to-state stability is equivalent to this form of stability.

Explicit Lyapunov functions are constructed for linear systems with unbounded input operators.

Abstract

We consider an abstract class of infinite-dimensional dynamical systems with inputs. For this class, the significance of noncoercive Lyapunov functions is analyzed. It is shown that the existence of such Lyapunov functions implies norm-to-integral input-to-state stability. This property in turn is equivalent to input-to-state stability, if the system satisfies certain mild regularity assumptions. For a particular class of linear systems with unbounded admissible input operators, explicit constructions of noncoercive Lyapunov functions are provided. The theory is applied to a heat equation with Dirichlet boundary conditions.