Description: The purpose of the article is to propose constructive methods for investigation the stability conditions of a linear hybrid automata. It is generally accepted to use the Lyapunov’s function method to design the stability of hybrid automata. The general theory of hybrid automata stability is rather complicated. This is due to the fact that unlike the theory of ordinary differential equations stability it is necessary to search for several Lyapunov’s functions for each individual state of the hybrid automaton. It is also necessary that all these functions be satisfied with a certain condition that ensures the stability of the solution. A constructive method for investigating the stability of a continuous-discrete process in the form of an optimization problem is constructed with the help of optimization methods. The theorems on the existence of a solution of the proposed problem and the criterion of its optimality are proved in the article. It is suggested to use a modified method of a generalized gradient to finding the optimal solution. For this purpose a generalized gradient is constructed and the rules for its calculation are given. Also, a computational procedure is presented.
Keywords: stability, optimal Lyapunov function, linear hybrid automaton, generalized gradient