A matrix-based approach to verifying stability and synthesizing optimal stabilizing controllers for finite-state automata

In this paper, we develop a matrix-based methodology to investigate the problems of stability and stabilizability for a deterministic finite automaton (DFA) in the framework of the semi-tensor product (STP) of matrices. First, we discuss the equilibrium point stability (resp., set stability) of a DFA, i.e., verifying whether or not all state trajectories starting from a subset of states converge to a specified equilibrium point (resp., subset of states). The necessary and sufficient conditions for verifying both stabilities are given, respectively. Second, equilibrium point stabilizability (resp., set stabilizability) of a DFA is investigated as verifying the issue of whether or not a DFA can be globally or locally stabilized to a specified equilibrium point (resp., subset of states) by a permissible state-feedback controller. Based on the pre-reachability set and invariant-subset defined in this paper, the matrix-based criteria for verifying equilibrium point stabilizability and set stabilizability are derived, respectively. Furthermore, for each type of stabilizability, all permissible state-feedback controllers for the case of minimal length state trajectories, called optimal state-feedback controllers, are characterized by using the proposed polynomial algorithms. Finally, two examples are presented to illustrate the effectiveness of the theoretical results.