[Paper Review] Input-to-state stability of distributed parameter systems

Andrii Mironchenko|arXiv (Cornell University)|Feb 1, 2023

Stability and Controllability of Differential Equations4 citations

TL;DR

This habilitation thesis establishes a comprehensive framework for input-to-state stability (ISS) in distributed parameter systems, including semilinear and boundary-controlled PDEs, using Lyapunov methods, small-gain theorems, and integral ISS criteria. It provides converse Lyapunov theorems for linear and semilinear systems, proves ISS for key equations like the heat and Kuramoto-Sivashinkiy equations, and develops non-coercive Lyapunov functions and small-gain conditions for infinite interconnections.

ABSTRACT

Input-to-state stability (ISS) allows estimating the impact of inputs and initial conditions on both the intermediate values and the asymptotic bound on the solutions. ISS has unified the input-output and Lyapunov stability theories and is a crucial property in the stability theory of control systems as well as for many applications whose dynamics depend on parameters, unknown perturbations, or other inputs. In this habilitation thesis, we provide a broad picture of infinite-dimensional input-to-state stability theory.

Motivation & Objective

• To develop a systematic theory of input-to-state stability (ISS) for infinite-dimensional systems governed by PDEs.
• To establish converse Lyapunov theorems for linear and semilinear evolution equations with unbounded input operators.
• To extend ISS analysis to boundary control systems and spatially distributed interconnections using small-gain techniques.
• To characterize integral ISS and develop Lyapunov-based criteria for nonlinear parabolic and higher-order PDEs.
• To provide rigorous stability conditions for specific equations such as the viscous Burgers’ and Kuramoto-Sivashinsky equations.

Proposed method

• Derives ISS criteria via non-coercive Lyapunov functions and dissipative forms for semilinear systems.
• Applies the density argument and mild solution theory to handle unbounded input operators in evolution equations.
• Uses spectral analysis and eigenfunction expansions to prove inequalities linking Sobolev and L^p norms.
• Establishes small-gain theorems in both max- and sum-formulations for infinite interconnections of ISS systems.
• Applies Agmon’s and Poincaré-type inequalities to bound state norms and derive stability conditions.
• Employs comparison functions and Dini derivatives to analyze stability properties and convergence rates.

Experimental results

Research questions

• RQ1 Under what conditions is a semilinear parabolic system with distributed or boundary inputs input-to-state stable?
• RQ2 Can a converse Lyapunov theorem be established for linear and semilinear distributed parameter systems with unbounded input operators?
• RQ3 How can small-gain theorems be extended to infinite interconnections of distributed parameter systems with nonlinear gains?
• RQ4 What Lyapunov function structures ensure integral ISS for nonlinear parabolic systems in Sobolev and L^p spaces?
• RQ5 How do spectral properties of differential operators influence the existence and form of ISS Lyapunov functions?

Key findings

• A converse Lyapunov theorem is established for semilinear systems, showing that ISS implies the existence of a non-coercive ISS Lyapunov function.
• The heat equation with Dirichlet boundary input is shown to be ISS using a Lyapunov function based on the H^1-norm.
• For the Kuramoto-Sivashinsky equation, a Lyapunov function is constructed using the inequality ∫x_zz² - λ∫x_z² ≥ σ(λ)∫x², with σ(λ) strictly decreasing and σ(4π²)=0.
• A small-gain theorem in max-formulation is proven for infinite interconnections of ISS systems, ensuring exponential ISS under a uniform small-gain condition.
• The thesis proves that linear systems with bounded input operators admit ISS Lyapunov functions, including the case P = -A⁻¹ for certain operators.
• For diagonal systems, ISS is characterized via spectral conditions on the eigenvalues, and a Lyapunov function is explicitly constructed.