On the Control of Discrete-Time Distributed Parameter Systems

A system described by a difference equation with a semigroup of operators is considered. The existence and uniqueness of an optimal control for the minimization of a quadratic cost functional over a finite interval is proved. The necessary condition for optimal control is derived and the optimal control is given by a linear state feedback law. The feedback operator is shown to be bounded, positive and self-adjoint, and the optimal cost is expressed in terms of the feedback operator. The control on the infinite interval and the behavior as $n \to \infty $ are considered.