Discrete-time jump linear systems with Markov chain in a general state space.

This thesis deals with discrete-time Markov jump linear systems (MJLS) with Markov chain in a general Borel space S. Several control issues have been addressed for this class of dynamic systems, including stochastic stability (SS), linear quadratic (LQ) optimal control synthesis, filter design and a separation principle.Necessary and sufficient conditions for SS have been derived.It was shown that SS is equivalent to the spectral radius of an operator being less than 1 or to the existence of a solution to a "Lyapunov-like" equation.Based on the SS concept, the finite-and infinite-horizon LQ optimal control problems were tackled.The solution to the finite-(infinite-)horizon LQ optimal control problem was derived from the associated control S-coupled Riccati difference (algebraic) equations.By S-coupled it is meant that the equations are coupled via an integral over a transition probability kernel having a density with respect to a σ-finite measure on the Borel space S. The design of linear Markov jump filters was analyzed and a solution to the finite-(infinite-)horizon filtering problem was obtained based on the associated filtering S-coupled Riccati difference (algebraic) equations.Conditions for the existence and uniqueness of a stabilizing positive semi-definite solution to the control and filtering Scoupled algebraic Riccati equations have also been derived.Finally a separation principle for discrete-time MJLS with Markov chain in a general state space was obtained.It was shown that the optimal controller for a partial information optimal control problem separates the partial information control problem into two problems, one associated with a filtering problem and the other associated with an optimal control problem with complete information.It is expected that the results obtained in this thesis may motivate further research on discrete-time MJLS with Markov chain in a general state space.