Optimal control of a time delay infinite order parabolic system with pointwise observation of the state

Various optimization problems for linear parabolic systems with multiple constant time delays are considered. In this paper, we consider an optimal distributed control problem for a linear infinite order parabolic system in which multiple constant time delays appear in the state equation. Sufficient conditions for the existence of a unique solution of the infinite order parabolic time delay equation with the Dirichlet boundary condition are proved. The time horizon T is fixed. Making use of the Lions scheme [6], 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.