The problem of optimal control for moving sources for systems with distributed parameters

A problem on optimal control of processes described by a set of equations of the parabolic type and an ordinary differential equation, with moving sources is investigated in the paper. For the considered problem of optimum control, the theorem of existence and uniqueness of the solution is proved, necessary conditions of optimality in the form of pointwise and integrated maximum principles are obtained, and sufficient conditions of Frechet differentiability of the criterion of quality are found and an expression for its gradient is obtained.