| In this work, we focus on two-time-scale systems with Markovian regime switching. The problems that we study are motivated by applications arising in optimization and control, wireless communication, financial engineering, signal processing, and related fields. Taking into consideration of various issues, the underlying Markov chain usually has a large state space, resulting in a large-scale system.; Treating large-scale systems, computational complexity is of practical concerns. We introduce a model involves multiple time scales. The time-scale separation then enables us to decompose the underlying system into subsystems and to aggregate the states in each subsystem into one state, which leads to substantial savings in terms of computational effort.; In this dissertation, after presenting some preliminaries, we consider near-optimal controls for a class of hybrid linear quadratic Regulator (LQ) problems with regime switching and with indefinite control weights in Part III, where the regime switching is modeled by a continuous-time Markov chain with a large state space. The use of large state space enables us to take various factors of uncertain environment into consideration, but it creates computational overhead and adds difficulties. Aiming at reduction of complexity, we demonstrate how to construct near-optimal controls. Introducing a small parameter to highlight the contrast of the weak and strong interactions and fast and slows motions results in a two-time-scale formulation. In view of the recent developments on LQ problems with indefinite control weights and two-time-scale Markov chains, We then establish the convergence of the system of Riccati equations associated with the hybrid LQ problem. Based on the optimal feedback control of the limit system obtained using the system of Riccati equations, we construct controls for the original problem and show that such controls are near-optimal. An example on Markowitz's mean-variance portfolio selection problem and a numerical demonstration of a simple system are also presented.; In Part IV of the dissertation, we develop a framework for balanced realizations of linear systems subject to regime switching. The regime switching is dictated by a continuous-time Markov chain with a finite state space. Precise definitions of balanced realizations are given. Then balanced realizations subject to regime switching is examined in detail. To reduce the complexity when the state space of the Markov chain is large, we use a two-time-scale formulation and decomposition/aggregation techniques to reduce the computational complexity. Approximation procedures are developed. Numerical examples are also presented for demonstration.; Finally, in Part V, a class of hybrid jump diffusions modulated by a Markov chain is considered. The motivation stems from insurance risk models, and emerging applications in production planning and wireless communications. Under suitable conditions, we develop asymptotic expansions of the transition densities for the underlying processes. We then validate the formal expansions by providing uniform error bounds. |
