Control óptimo no convexo de ecuaciones diferenciales elípticas con restricciones puntuales de estado.

Abstract

We consider an optimal control problem governed by an elliptic partial differential equation with homogeneous Neumann conditions, whose cost function involves a non-convex penalty term and pointwise state constraints are imposed. The usual tools from convex analysis cannot be applied directly for deriving an optimality system therefore, it is necessary to introduce techniques from nonsmooth analysis. In addition, the presence of pointwise state constraints leads to poor regularity Lagrange multipliers, making it difficult to compute solutions numerically. Due to these difficulties, we introduce two regularizations to deal with the lack of convexity of the cost function and the low regularity of the Lagrange multipliers associated with the pointwise state constraints. Specifically, we use a Huber-like regularization to cope with the quasinorm, whereas a Lavrentiev regularization is considered for the state constraints. Thanks to these regularizations, it is possible to formulate a family of optimal control problems (depending on the regularization parameters) for which we derive an optimality system by using the Difference of Convex Functions (DC) theory involving more regular Lagrange multipliers.

Later, we solve the regularized problem numerically by applying the semi-smooth Newton method. We analyze the qualitative properties of the solution of the regularized problem, and we numerically asses the variation of the parameters associated to the regularized problem. Finally, we study the convergence rate of the proposed algorithm numerically, and we draw conclusions and discuss open questions for future research.