An optimal control problem by a hyperbolic system with boundary delay

The paper deals with an optimal control problem by a system of semilinear hyperbolic equations with boundary differential conditions with delay. This problem is considered for smooth controls. Because this requirement it is impossible to prove optimality conditions of Pontryagin maximum principle type and classic optimality conditions of gradient type. Problems of this kind arise when modeling the dynamics of non-interacting age-structured populations. Independent variables in this case are the age of the individuals and the time during which the process is considered. The functions of the process state describe the age-related population density. The goal of the control problem may be to achieve the specified population densities at the end of the process. The problem of identifying the functional parameters of models can also be considered as the optimal control problem with a quadratic cost functional. For the problem we obtain a non-classic necessary optimality condition which is based on using a special control variation that provides smoothness of controls. An iterative method for improving admissible controls is developed. An illustrative example demonstrates the effectiveness of the proposed approach.