  This paper mainly investigates the optimal control and stabilization problems
for linear discrete-time Markov jump systems. The general case for the
finite-horizon optimal controller is considered, where the input weighting
matrix in the performance index is just required to be positive semi-definite.
The necessary and sufficient condition for the existence of the optimal
controller in finite-horizon is given explicitly from a set of coupled
difference Riccati equations (CDRE). One of the key techniques is to solve the
forward and backward stochastic difference equation (FDSDE) which is obtained
by the maximum principle. As to the infinite-horizon case, we establish the
necessary and sufficient condition to stabilize the Markov jump linear system
in the mean square sense. It is shown that the Markov jump linear system is
stabilizable under the optimal controller if and only if the associated couple
algebraic Riccati equation (CARE) has a unique positive solution. Meanwhile,
the optimal controller and optimal cost function in infinite-horizon case are
expressed explicitly.
