Input-to-state stability of infinite-dimensional systems: Foundations and present-day developments

TL;DR

This work surveys input-to-state stability (ISS) for infinite-dimensional systems, unifying stability and robustness for distributed-parameter dynamics via Lyapunov methods and small-gain analysis. It covers direct and converse ISS Lyapunov theorems, semigroup/admissibility techniques for systems with unbounded input operators, and nonlinear small-gain results for interconnections, highlighting both coercive and non-coercive Lyapunov frameworks. The paper demonstrates ISS for linear and nonlinear PDEs, with illustrative examples in parabolic and hyperbolic settings, including boundary control and interconnection networks. By providing tractable dissipativity criteria and explicit interconnection conditions, it offers a practical toolkit for robust stability analysis and controller design in distributed-parameter systems and networked PDEs. The work thus underpins robust control design for complex PDE-based applications such as tokamak dynamics, traffic flow, and multi-physics networks.

Abstract

Input-to-state stability (ISS) unifies the stability and robustness in one notion, and serves as a basis for broad areas of nonlinear control theory. In this contribution, we covered the most fundamental facts in the infinite-dimensional ISS theory with a stress on Lyapunov methods. We consider various applications given by different classes of infinite-dimensional systems. Finally, we discuss a Lyapunov-based small-gain theorem for stability analysis of an interconnection of two ISS systems.

Figures (1)

• Figure 1: Typical solution of an ISS system with $u\not\equiv 0$

Theorems & Definitions (14)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Theorem 1: ISS superposition theorem, MiW18b
• Definition 5
• Theorem 2
• Theorem 3
• Theorem 4
• Definition 6