Weak asymptotic stabilizability of discrete-time systems given by set-valued operators
Let \(T\) be a convex process in a Banach space \(X\) (i.e. \(T\) is a multioperator whose graph is a closed convex cone). A discrete inclusion \[ x(k+1)\in T(x(k)), \qquad k=0,1,\dots \] is said to be weakly asymptotically stabilizable if for every \(x_0\in X\) there is a solution \(x(k)\) such that \(x(0)=x_0\) and \(x(k)\to0\) as \(k\to\infty\). The author characterizes spectral properties of closed convex multioperators in Banach space. On that base, sufficient conditions for weak asymptotic stabilizability of discrete inclusions in terms of controllability-like conditions are given. The results are applied to the study of linear discrete control systems of the form \[ x(k+1)= Ax(k)+Bu(k), \qquad u(k)\in\Omega. \]
scientific article; zbMATH DE number 930196