Sensitivity analysis of time delay parabolic-hyperbolic optimal control problems with boundary conditions involving time delays

In the paper the first order sensitivity analysis is performed for a class of optimal control problems for parabolic-hyperbolic systems in which time delays appear both in the state equations and in the Neumann boundary conditions. A singular perturbation of geometrical domain of integration is introduced in the form of a circular hole. The Steklov-Poincaré operator on a circle is defined in order to reduce the problem to regular perturbations in the truncated domain. The optimality system is differentiated with respect to the small parameter and the directional derivative of the optimal control is obtained as a solution to an auxiliary optimal control problem.