On the Issue of Stability of Hybrid Automata by a Part of Variables

Abstract

The problem of the stability of hybrid automata according to certain variables is considered. This problem is urgent and develops rapidly, especially in recent years. Aspects of its solution using the Lyapunov functions are also considered. The problem of hybrid automata stability regarding certain variables arises naturally for solving the applied problems. Namely, when based on the requirements of the normal functioning of an object, it is sufficient to ensure its stability only according to some variables. Formulation of the problem of stability regarding certain variables belongs to A.M. Lyapunov, but he himself did not investigate this problem. There is a great methodological similarity in the study of stability considering all, and a part of variables using the Lyapunov functions. However, there are certain differences in resolving some identical issues as applied to stability problems for all and a part of the variables. There are methods to reduce the problem of stability regarding certain variables to the study of stability in all variables of some auxiliary system, and vice versa. These two types of stability are closely related and mutually complementary. Currently, the problem of the stability of hybrid automata in terms of variables is considered as an independent section of the theory of stability. It is shown that the property of the y1-positive definiteness of Lyapunov functions is not sufficient for studying the stability of hybrid automata in terms of a part of variables. The concept of a y1-uniform positive definiteness of a function had been introduced. Theorems that provide sufficient stability conditions had been proved. For linear hybrid automata the constructive stability conditions had been obtained. The article also shows how using the stated theorems one can investigate the stability of hybrid timed automata.