Optimal control of a time delay hyperbolic system with the Neumann boundary condition

In this paper, we consider an optimal distributed control problem for a linear 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 hyperbolic time delay equation with the Neumann boundary condition are proved. The time horizon T is fixed. Making use of the Lions scheme [8], 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.