Pontryagin’s maximum principle for a mixed-constrained optimal control problem governed by the heat equation

The article investigates the problem of optimal control of a system of partial differential equations of second-order parabolic type in the case of mixed constraints. For this problem, a necessary optimality condition is formulated in the form of the Pontryagin’s maximum principle. This result can be useful both for organizing a subsequent computational procedure such as the method of successive approximations, and for a qualitative analysis of the optimization problem, which may not lead to a final answer, but establishes important properties of the solution, that is, the optimal process. As an example of the application of the maximum principle, the problem of optimal control of the heat equation is considered. In both problems, the controlled system with distributed parameters is singular according to Zh.L. Lyons. In the case of a singular system, the application of the classical theory of optimal control is either difficult or impossible. The presence of mixed constraints in the formulation of the problems under consideration significantly complicates the process of finding the optimal process.