Lower semicontinuity of the solution map to a parametric elliptic optimal control problem with mixed pointwise constraints

Abstract

This paper studies the solution stability of a parametric optimal control problem governed by single linear elliptic equations with mixed control-state constraints and convex cost functions. By reducing the problem to a parametric programming problem and a parametric variational inequality, we obtain sufficient conditions under which the solution map to an elliptic optimal control problem is lower semicontinuous.

Keywords:

• parametric optimal control
• solution stability
• lower semicontinuity
• elliptic equation

AMS Subject Classifications:

• 49K20
• 93B50
• 93C05
• 93C20
• 93C35
• 76D05
• 65J15

Acknowledgments

The authors wish to express their thanks to the anonymous referees for their helpful suggestions and comments which improved the original manuscript greatly. The first author wishes to thank the University of Duisburg-Essen and the Alexander von Humboldt Foundation for hospitality and financial support.