On a class of optimal control problems with distributed and lumped parameters

The optimal control of moving sources governed by a parabolic equation and a system of ordinary differential equations with initial and boundary conditions is considered. For this problem, an existence and uniqueness theorem is proved, sufficient conditions for the Fréchet differentiability of the cost functional are established, an expression for its gradient is derived, and necessary optimality conditions in the form of pointwise and integral maximum principles are obtained.