Input-to-state stability of infinite-dimensional control systems

Abstract

We develop tools for investigation of input-to-state stability (ISS) of infinite-dimensional control systems. We show that for certain classes of admissible inputs the existence of an ISS-Lyapunov function implies the input-to-state stability of a system. Then for the case of systems described by abstract equations in Banach spaces we develop two methods of construction of local and global ISS-Lyapunov functions. We prove a linearization principle that allows a construction of a local ISS-Lyapunov function for a system which linear approximation is ISS. In order to study interconnections of nonlinear infinite-dimensional systems, we generalize the small-gain theorem to the case of infinite-dimensional systems and provide a way to construct an ISS-Lyapunov function for an entire interconnection, if ISS-Lyapunov functions for subsystems are known and the small-gain condition is satisfied. We illustrate the theory on examples of linear and semilinear reaction-diffusion equations.Comment: 33 page

Robust stabilization of delayed neural fields with partial measurement and actuation

“…After having extended the classical Lyapunov sufficient condition for ISS (Sontag, 1989), we present a small-gain theorem applicable to such class of systems (Section 3). This extension is in line with recent works addressing ISS for infinite-dimensional systems (Dashkovskiy and Mironchenko, 2013;Karafyllis and Jiang, 2007;Mironchenko and Ito, 2016;Prieur and Mazenc, 2012), including retarded functional differential equations (Pepe and Jiang, 2006;Karafyllis et al, 2008;Mazenc et al, 2008;Teel, 1998). Focusing on delayed neural fields under partial proportional feedback, we then show that the uncontrolled population is ISS with respect to the state of the controlled population and possible exogenous signals, provided that the spatial L 2 -norm of its internal synaptic weights is below a certain bound.…”

Construction of Lyapunov Functions for Interconnected Parabolic Systems: An iISS Approach

“…The purpose of this section is to develop a Lyapunov-type characterization of ISS for PDEs in (1.2). There is a number of papers, where such characterizations for parabolic systems whose state space is an L p space have been provided [24,4]. However, as we will see in Section 7 the iISS systems in many cases cannot have the L p space both as an input and state space.…”

“…Many classes of evolution PDEs, such as parabolic and hyperbolic equations are of this kind [9], [3]. As in the case of finite-dimensional systems [29], the notion of an ISS Lyapunov function can be defined for (1.1) so that the existence of an ISS Lyapunov function is sufficient for ISS of (1.1) (see [4]). This motivated the results in [4] on constructions of ISS Lyapunov functions for a class of parabolic systems belonging to (1.1).…”

“…In [4,5], ISS of large scale systems whose subsystems are in the form of (1.1) has been studied and the ISS small gain theorem, already available for finite-dimensional systems (see [17,6]) has been extended to the infinite-dimensional systems. However, the method does not accommodate iISS subsystems which are not ISS.…”

“…The primary goal of this paper is to accomplish an iISS small gain theorem [13,11], originally proved for finite-dimensional systems, in the infinite-dimensional setting. In contrast to the small-gain theorem from [4], we require ISS property only from one subsystem and not from both of them; the other subsystem may be only iISS.…”