Matrix approach to stabilizability of deterministic finite automata

This paper investigates the state feedback stabilizing problem of deterministic finite automata using a matrix approach. With the help of semi-tensor product, a matrix-based expression for finite automata is given, and the dynamics of automata are expressed in the form of a discrete-time bilinear equation. After providing the notions of equilibrium and cycle stability, we give necessary and sufficient algebraic conditions for the stabilizability of deterministic finite automata for the two respective cases. Then, based on the matrix expression, we focus on a special case where the controlled state trajectories to the target equilibrium is minimal. All the state feedback controllers can be obtained by solving a matrix inequality. Examples are also given for illustration.