Optimal Control of a Time Delay Infinite Order Hyperbolic System with Pointwise Observation of the State 1

In this paper, we consider an optimal distributed control problem for a linear infinite order hyperbolic system in which time delay given in the integral form appears in the state equation. Sufficient conditions for the existence of a unique solution of the infinite order hyperbolic time delay equation with the Dirichlet boundary condition are proved. The time horizon T is fixed. Making use of the Lions scheme (Lions [1971]), necessary and sufficient conditions of optimality for the Dirichlet problem with the quadratic performance functional with pointwise observation of the state and constrained control are derived.