Characterizations of Input-to-State Stability for Infinite-Dimensional Systems

• Published in: IEEE transactions on automatic control Vol. 63; no. 6; pp. 1692 - 1707
• Main Authors: Mironchenko, Andrii, Wirth, Fabian
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.06.2018; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Asymptotic stability; Banach spaces; Control stability; Control systems; Criteria; Differential equations; Dimensional stability; Infinite-dimensional systems; input-to-state stability (ISS); Liapunov functions; Lyapunov methods; Mathematical analysis; nonlinear systems; Ordinary differential equations; Stability criteria; Thermal stability; Time delay systems
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2017.2756341

We prove characterizations of input-to-state stability (ISS) for a large class of infinite-dimensional control systems, including some classes of evolution equations over Banach spaces, time-delay systems, ordinary differential equations (ODE), and switched systems. These characterizations generalize well-known criteria of ISS, proved by Sontag and Wang for ODE systems. For the special case of differential equations in Banach spaces, we prove even broader criteria for ISS and apply these results to show that (under some mild restrictions) the existence of a noncoercive ISS Lyapunov functions implies ISS. We introduce the new notion of strong ISS (sISS) that is equivalent to ISS in the ODE case, but is strictly weaker than ISS in the infinite-dimensional setting and prove several criteria for the sISS property. At the same time, we show by means of counterexamples that many characterizations, which are valid in the ODE case, are not true for general infinite-dimensional systems.