Non-differentiable optimal control problems of retarded parabolic systems

Various optimization problems associated with the optimal control of distributed-parameter systems with time delays appearing in the boundary conditions have been studied recently in [1]-[8] and [11], [12]. In this paper, we consider an optimal boundary control problem for a linear parabolic system with deviating argument given in the integral form. Sufficient conditions for the existence of a unique solution of the parabolic lag equation with the Neumann boundary condition involving a retarded argument in the integral form with h ε (0,6) are presented. The cost function constitutes the sum of a differentiable and non-differentiable function. The time horizon T is fixed. Finally, we impose some constraints on the control. Making use of the Lions scheme [9], necessary and sufficient conditions of optimality for the Neumann problem are derived.