Lipschitz continuity of solutions and corresponding multipliers to distributed and boundary semilinear elliptic optimal control problems with mixed pointwise control-state constraints

TL;DR

This paper investigates the existence, regularity, and Lipschitz continuity of solutions and multipliers in semilinear elliptic optimal control problems with mixed pointwise control-state constraints, acting in the domain and on the boundary.

Contribution

It establishes the existence and Lipschitz regularity of minimizers and multipliers for a broad class of elliptic control problems with mixed constraints.

Findings

Existence of minimizers and multipliers under general assumptions

Lipschitz continuity of optimal solutions and multipliers

Development of Sobolev space tools for fractional order functions

Abstract

This paper is concerned with the existence and regularity of mininizers as well as of corresponding multipliers to an optimal control problem governed by semilinear elliptic equations, in which mixed pointwise control-state constraints are considered in a quite general form and the controls act simultaneously in the domain and on the boundary. Under standing assumptions, the minimizers and the corresponding multipliers do exist. Furthermore, by applying the bootstrapping technique and establishing some calculation tools for functions in Sobolev spaces of fractional order, the optimal solutions and the associated Lagrange multipliers are shown to be Lipschitz continuous.