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), switched systems. These characterizations generalize wellknown 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 non-coercive ISS Lyapunov functions implies ISS. We introduce the new notion of strong ISS which is equivalent to ISS in the ODE case, but which 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.Index Terms-input-to-state stability, nonlinear systems, infinite-dimensional systems.
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
Abstract. We prove that impulsive systems, which possess an ISS Lyapunov function, are ISS for time sequences satisfying the fixed dwell-time condition. If an ISS Lyapunov function is the exponential one, we provide a stronger result, which guarantees uniform ISS of the whole system over sequences satisfying the generalized average dwell-time condition. Then we prove two smallgain theorems that provide a construction of an ISS Lyapunov function for an interconnection of impulsive systems, if the ISS-Lyapunov functions for subsystems are known. The construction of local ISS Lyapunov functions via linearization method is provided. Relations between small-gain and dwell-time conditions as well as between different types of dwell-time conditions are also investigated. Although our results are novel already in the context of finite-dimensional systems, we prove them for systems based on differential equations in Banach spaces that makes obtained results considerably more general.
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.
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