Pointwise observation of the state given by the parabolic system with the Neumann boundary condition

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 parabolic system in which multiple constant time delays appear in the state equation. Sufficient conditions for the existence of a unique solution of the parabolic time delay equation with the Neumann boundary condition are proved. The time horizon T is fixed. Making use of the Lions scheme [10], necessary and sufficient conditions of optimality for the Neumann problem with the quadratic performance functional with pointwise observation of the state and constrained control are derived. The example of application is also presented.