  This paper deals with the finite horizon optimal control problem for
discrete-time Markov jump linear system with input delay. The correlation among
the jumping parameters and the input delay are considered simultaneously, which
forms the basic difficulty of the design. one of the key techniques is to solve
a delayed forward and backward jumping parameter difference equation which is
obtained by an improved maximum principle, and the other is the introduction of
a "d-step backward formula". Based on the proposed techniques, a necessary and
sufficient condition for the existence of the optimal controller is given in an
explicit form and an analytical solution to the optimal controller is supplied.
The optimal controller is a linear function of the current time state and the
historical time control input, where the feedback gains are a set of jumping
parameter matrices derived by solving a new type of coupled difference Riccati
equation. The key step in the derivation is to establish the relationship
between the costate and the real state of the system. The result obtained in
this paper can be viewed as a generalization of the standard case, in which
there is only one mode of operation.
