  This paper will investigate the infinite horizon optimal control and
stabilization problems for the Markov jump linear system (MJLS) subject to
control input delay. Different from previous works, for the first time, the
necessary and sufficient stabilization conditions are explored under explicit
expressions, and the optimal controller for infinite horizon is designed with a
coupled algebraic Riccati equation. By introducing a new type of Lyapunov
equation, we show that under the exact observability assumption, the MJLS with
control input delay is stabilizable in the mean square sense with the optimal
controller if and only if a coupled algebraic Riccati equation has a unique
positive definite solution. The presented results are parallel to the optimal
control and stabilization for standard system with input delay.
