[Paper Review] Noncoercive Lyapunov functions for input-to-state stability of infinite-dimensional systems

Birgit Jacob, Andrii Mironchenko+2 more|arXiv (Cornell University)|Nov 4, 2019

Stability and Controllability of Differential EquationsEngineering44 references19 citations

TL;DR

This paper establishes that the existence of noncoercive Lyapunov functions implies norm-to-integral input-to-state stability (ISS) for a broad class of infinite-dimensional systems, and under mild regularity conditions, this implies full ISS. The authors provide explicit constructions of such Lyapunov functions for linear systems with unbounded input operators, including a heat equation with Dirichlet boundary control, marking the first such construction for this system using Lyapunov methods.

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.

Motivation & Objective

• To investigate whether noncoercive Lyapunov functions—common in infinite-dimensional systems—can imply input-to-state stability (ISS).
• To extend existing ISS characterizations to systems with unbounded input operators, including boundary control systems.
• To provide explicit constructions of noncoercive ISS Lyapunov functions for linear systems with $\infty$-admissible input operators.
• To demonstrate the applicability of the theory by constructing a Lyapunov function for a heat equation with Dirichlet boundary control.

Proposed method

• The authors define a general class of forward-complete infinite-dimensional control systems with inputs, including those governed by semigroups.
• They introduce the concept of norm-to-integral ISS, a weaker stability notion that links to noncoercive Lyapunov functions.
• Using the resolvent of the infinitesimal generator at zero as a candidate operator $P$, they construct a Lyapunov function $V(x) = -\langle A^{-1}x, x\rangle_X$.
• They prove that if this $V$ is a noncoercive Lyapunov function, then the system is norm-to-integral ISS.
• They establish equivalence between norm-to-integral ISS and full ISS under mild regularity assumptions: continuity of the flow near the origin and boundedness of finite-time reachability sets.
• They apply the theory to a one-dimensional heat equation with Dirichlet boundary control, verifying the ISS estimates and constructing the Lyapunov function explicitly.

Experimental results

Research questions

• RQ1 Can noncoercive Lyapunov functions imply input-to-state stability in infinite-dimensional systems, especially when coercivity fails?
• RQ2 Under what conditions does norm-to-integral ISS imply full input-to-state stability?
• RQ3 How can noncoercive Lyapunov functions be explicitly constructed for linear systems with unbounded input operators?
• RQ4 Is it possible to construct an ISS Lyapunov function for a heat equation with Dirichlet boundary control using this framework?
• RQ5 What is the role of the resolvent at zero in constructing such Lyapunov functions for analytic semigroups?

Key findings

• The existence of a noncoercive Lyapunov function implies norm-to-integral ISS for a broad class of infinite-dimensional systems.
• Norm-to-integral ISS is equivalent to full ISS if the system's flow is continuous near the origin and finite-time reachability sets are bounded.
• For linear systems with $\infty$-admissible input operators, the Lyapunov function $V(x) = -\langle A^{-1}x, x\rangle_X$ is noncoercive and implies ISS under suitable spectral conditions on the generator $A$.
• The constructed Lyapunov function for the one-dimensional heat equation with Dirichlet boundary control is the first such explicit construction using Lyapunov methods.
• The paper provides a systematic framework for constructing noncoercive ISS Lyapunov functions for analytic systems with subnormal generators.
• The method applies to systems where traditional coercive Lyapunov functions fail, offering a viable alternative for stability analysis in infinite dimensions.