Asymptotical Controllability of Continuous-Time Probabilistic Logic Control Systems

Herein, we investigate the asymptotical controllability of continuous-time probabilistic logic control systems (CT-PLCSs) under piecewise constant and right continuous controls (PCRCCs). The theory of an invariant subset for a CT-PLCS controlled by a PCRCC is developed. Based on this theory, we prove that a state is asymptotically reachable from another state under a PCRCC if and only if it is asymptotically reachable under a simple periodic PCRCC; in other words, the control efforts are pairwise distinct within one period. Thereafter, a verifiable, necessary and sufficient condition for the asymptotical reachability between states is obtained, based on which verifiable, necessary and sufficient conditions for the asymptotical reachability and asymptotical controllability are derived. In particular, we prove that a CT-PLCS is asymptotically controllable if and only if each state can be asymptotically stabilized by PCRCCs. Additionally, the proof of the proposed results is constructive, which naturally leads to the development of methods of designing PCRCCs. Lastly, examples are provided to illustrate the theoretical results.