Language stability and stabilizability of discrete event dynamical systems

This paper studies the stability and stabilizability of discrete event dynamical systems (DEDSs) modeled by state machines. Stability and stabilizability are defined in terms of the behavior of the DEDSs, i.e. the language generated by the state machines (SMs). This generalizes earlier work where they were defined in terms of legal and illegal states rather than strings. The notion of reversal of languages is used to obtain algorithms for determining the stability and stabilizability of a given system. The notion of stability is then generalized to define the stability of infinite or sequential behavior of a DEDS modeled by a Büchi automaton. The relationship between the stability of finite and stability of infinite behavior is obtained and a test for stability of infinite behavior is obtained in terms of the test for stability of finite behavior. An algorithm of linear complexity for computing the regions of attraction is presented, which is used for determining the stability and stabilizability of a given system defined in terms of legal states. This algorithm is then used to obtain efficient tests for checking sufficient conditions for language stability and stabilizability.