Abstract:
This article is concerned with an optimal control problem governed by a Laplace equation. Initially, the optimal control problem, governed by a system of partial differential elliptic equations of the second order, is considered. The case of a system, that is singular according to Lions, is considered. In this system a given control may give rise to either no of any state or, on the contrary, the infinitely many ones or that to a single but unstable state [1]. In this situation, the application of the classic optimal control theory is either very difficult or impossible. Special methods applicable to the control problems, governed by singular distributed systems, are developed in the works of Zh. L. Lions, I. Ekland, P. Marselini, G. Mossino, P. Rivera, and of many other authors. But it should be noted that in most of these works the simplest problem statement is discussed. It is defined by the fact that the set of admissible processes, i.e., the processes, among which we seek the minimum of certain functional, is described by a differential equation and the connected with it, boundary conditions only. In the present work a more general and complex case is considered, namely, the case that in the description of the above-mentioned set there are so-called state constraints. This implies that the phase vector of a system does not leave the given set. In such a statement the optimal control problem, governed by a distributed singular system, is, undoubtedly, of substantial interest. Next it will be shown that the optimal process in this problem is generated by a nonlinear optimal controller and its equation will be obtained.
Keywords:maximum principle of Pontryagin's type, Laplace equation, optimal controller.
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