Weak asymptotic stability at first approximation for periodic differential inclusions
This paper represents a continuation of very interesting research by the author on ``first'' (in some sense linear) approximations to study the stability behavior of differential inclusions and control systems [see also the author, Autom. Remote Control 51, No. 7, 901-908 (1990; Zbl 0737.93068); ibid. No. 8, 1052-1058 (1990; Zbl 0779.93087)]. The concept of weak asymptotic stability for generalized control systems was established in the 1960's and has gained in importance more recently in the context of differential inclusions. In the above-mentioned papers by the same author, this concept has been linked with the one of a ``first approximation'' to an autonomous control system or differential inclusion. Here, the same type of results are extended to non-autonomous periodic systems, and it is interesting that this link is established considering the intermediate step of discrete-time inclusions, which therefore are also included in these investigations. The techniques used in establishing and phrasing these results are used in the ``set-valued analysis'' and recent research in differential inclusions. To give the flavor of them I give an abbreviated statement of theorem 4.1. Consider a differential inclusion (1) \(\dot x\in F(x,t)\), periodic in \(t\), and \(A(x, t)\) the first approximation of the map \(x\to \text{co } F(x, t)\), both satisfying precisely given (and quite reasonable) conditions. Then the following statements are equivalent: a) the equilibrium position \(x= 0\) of differential inclusion (1) is weakly asymptotically stable; b) the ``adjoint differential inclusion'' \(\dot x^*\in - A^*(x^*, t)\) has only the trivial solution with nonnegative Lyapunov exponent; c) the eigenvalues of the corresponding discretized system \(A^*\), restricted to the maximal invariant subspace, have absolute values less than one. The results are finally also translated into the setting of the usual control theory (parametrized by a control \(u(t)\)).
scientific article; zbMATH DE number 822779
scientific article