Necessary Conditions for Optimal Controls of Elliptic or Parabolic Problems

Abstract

In this paper I prove necessary conditions for optimal controls of boundary value parabolic or elliptic problems, when the coefficients of the operators depend on the controls. Some stochastic control problems can be formulated as optimal controls for the coefficients of elliptic or parabolic problems, and the necessary conditions proved here generalize known results to the case when the second order coefficients depend on the controls. From the necessary conditions uniqueness criteria and “bang-bang” theorems are deduced, and, suggested by the direct physical interpretation of these problems, stability (in a variational sense) theorems are proved.