Averaging, aggregation and optimal control of singularly perturbed stochastic hybrid systems

This paper develops methodologies for the averaging, aggregation and optimal control of stochastic hybrid systems whose state equations depend on continuous-time, nearly completely decomposable finite state Markov chains. Such Markovian switching models are assumed to have several identified groups of strongly interacting states. The random sample solution process of the system state can be well approximated by a deterministic trajectory, for the duration of an interval of which the switching process sojourns in a group. Aggregated models are obtained by utilizing the aggregation method over the infinite time interval. By using the perturbation approach, necessary and sufficient conditions of stochastic stabilizability and controllability proposed by Ji and Chizeck (1990) are presented for the systems. With the aggregation models and the known results of stochastic stabilizability, near-optimum finite time and infinite time Markovian jump linear quadratic control laws are also investigated. The limiting behaviour of corresponding minimized averaged cost functions is studied. Finally, an illustrative example is performed to illustrate the feasibility and effectiveness of the proposed techniques.