[Paper Review] Input-to-state stability of infinite-dimensional systems: recent results and open questions

Andrii Mironchenko, Christophe Prieur|arXiv (Cornell University)|Oct 3, 2019

Stability and Controllability of Differential Equations4 citations

TL;DR

This survey provides a comprehensive overview of input-to-state stability (ISS) for infinite-dimensional systems, unifying Lyapunov and input-output methods to analyze robustness against disturbances and initial conditions. It establishes foundational theory for linear and nonlinear PDEs, boundary control systems, time-delay systems, and interconnected networks, offering tools for stability analysis and control design via Lyapunov functions and small-gain theorems.

ABSTRACT

In a pedagogical but exhaustive manner, this survey reviews the main results on input-to-state stability (ISS) for infinite-dimensional systems. This property 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 paper, starting from classic results for nonlinear ordinary differential equations, we motivate the study of ISS property for distributed parameter systems. Then fundamental properties are given, as an ISS superposition theorem and characterizations of (global and local) ISS in terms of Lyapunov functions. We explain in detail the functional-analytic approach to ISS theory of linear systems with unbounded input operators, with special attention devoted to ISS theory of boundary control systems. The Lyapunov method is shown to be very useful for both linear and nonlinear models, including parabolic and hyperbolic partial differential equations. Next, we show the efficiency of the ISS framework to study the stability of large-scale networks, coupled either via the boundary or via the interior of the spatial domain. ISS methodology allows reducing the stability analysis of complex networks, by considering the stability properties of its components and the interconnection structure between the subsystems. An extra section is devoted to ISS theory of time-delay systems with the emphasis on techniques, which are particularly suited for this class of systems. Finally, numerous applications are considered in this survey, where ISS properties play a crucial role in their study. This survey suggests many open problems throughout the paper.

Motivation & Objective

• To unify Lyapunov and input-output stability theories for infinite-dimensional systems using ISS as a central framework.
• To extend ISS theory from finite-dimensional ODEs to distributed parameter systems governed by PDEs and time-delay equations.
• To develop and systematize Lyapunov-based methods for analyzing stability of linear and nonlinear PDE systems with unbounded input operators.
• To apply ISS to interconnected systems, including networks of PDEs coupled via boundaries or interiors, using small-gain theorems.
• To identify and highlight open problems in ISS theory for infinite-dimensional systems, particularly in boundary control and time-delay systems.

Proposed method

• Utilizes the ISS superposition theorem to characterize ISS as a combination of global asymptotic stability and global attractivity under inputs.
• Applies Lyapunov functions to prove ISS for linear systems with unbounded input operators, especially in boundary control settings.
• Employs functional-analytic techniques to study well-posedness and stability of evolution equations in Hilbert and Banach spaces.
• Applies the nonlinear small-gain theorem to analyze stability of large-scale networks of infinite-dimensional systems.
• Adapts ISS theory to time-delay systems using specialized Lyapunov-Krasovskii functionals and delay-dependent analysis.
• Integrates ISS with robust control methods, including backstepping and feedback redesign, for PDEs with uncertainties.

Experimental results

Research questions

• RQ1 How can ISS be characterized for infinite-dimensional systems using Lyapunov functions, especially for systems with unbounded input operators?
• RQ2 What are the necessary and sufficient conditions for ISS in linear and nonlinear PDE systems, including parabolic and hyperbolic equations?
• RQ3 How can ISS be applied to analyze the stability of interconnected infinite-dimensional systems, such as networks of PDEs coupled via boundaries or interiors?
• RQ4 What are the key challenges and open problems in extending ISS theory to boundary control systems and time-delay systems?
• RQ5 In what ways does ISS unify and generalize finite-dimensional stability concepts in the context of distributed parameter systems?

Key findings

• ISS is equivalent to the existence of a smooth ISS Lyapunov function, generalizing classical theorems of Massera and Kurzweil to systems with inputs.
• The ISS superposition theorem characterizes ISS as the combination of global asymptotic stability in the absence of inputs and a global attractivity property under inputs.
• For linear systems with unbounded input operators, ISS can be analyzed using functional-analytic methods and spectral properties of the generator.
• Boundary control systems can be proven ISS using Lyapunov-based techniques that account for the unbounded nature of the input operator.
• The nonlinear small-gain theorem enables stability analysis of large-scale networks by combining component-wise ISS properties and interconnection structure.
• Time-delay systems can be effectively analyzed using ISS frameworks tailored to delay-dependent dynamics, with Lyapunov-Krasovskii functionals providing sufficient conditions for stability.