Optimal control by a cascade system of hyperbolic and ordinary delayed differential equation

In the class of smooth control functions, an optimal control problem of first-order semilinear hyperbolic equations is investigated. We consider the case when the functional parameter in the right side of the hyperbolic system is determined from the controlled system of ordinary differential equations with constant state delay. Control functions are restricted by pointwise (amplitude) constraints. Problems of this kind arise when modeling a number of processes of population dynamics, interaction of a fluid (liquid or gas) with solids, etc. Optimal control methods based on the use of the Pontryagin maximum principle, its consequences and modifications are not applicable for such problems. The proposed approach is based on a special control variation, which ensures the smoothness of variable controls and the fulfillment of restrictions. The necessary optimality condition is proved. A scheme of a method for improving permissible control based on this condition is proposed, the convergence of the method is justified. An illustrative example is given.