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

Various optimization problems associated with the optimal control of distributed hyperbolic systems with time delays appearing in the boundary conditions have been studied recently in Refs. [2]-[7] respectively. In this paper, we consider an optimal control problem for a linear hyperbolic system in which multiple time delays appear in the state equation. The initial conditions are given by control functions. Sufficient conditions for the existence of a unique solution of such hyperbolic equations with the Neumann boundary conditions are presented. The performance functional has the quadratic form. The time horizon T is fixed. Finally, we impose some constraints on the control. Making use of the Lions scheme [8], necessary and sufficient conditions of optimality for the Neumann problem with the quadratic performance functional and constrained control are derived.