Averaging, aggregation and optimal control of stochastic hybrid systems with singularly perturbed Markovian switching behavior

This paper develops methodologies for the averaging, aggregation and optimal control of stochastic hybrid systems whose state equations depend on continuous-time and nearly completely decomposable finite state Markov chains. Such Markovian switching models consist of several identified groups of strongly interacting states. The random sample solution processes of the system state can be well approximated by deterministic trajectories, for the duration of intervals of which the switching process takes values in different groups. Aggregated models are derived, by utilizing averaging and aggregation ideas, to describe the global behavior for the original systems. By modifying concepts of stochastic stabilizability and controllability, necessary and sufficient conditions are established by the perturbation approach. Based on these properties, near-optimum finite and infinite time Markovian jump linear quadratic control problems are explored as well and corresponding averaged cost functions are studied. All the results are shown to hold when the system state and the group index can be exactly measured. Finally, an illustrative example is provided to demonstrate the aforementioned techniques.