Optimal control via initial conditions of a time delay hyperbolic system

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 Dirichlet 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 Dirichlet problem with the quadratic performance functional and constrained control are derived.