On an optimal control problem with gradient constraints

ABSTRACT

Usually, control functions in control-constrained optimal control are chosen from a Lebesgue space. This choice, however, makes it impossible to postulate additional conditions on the control function's slope which is practically relevant in some situations. In order to overcome this disadvantage, a natural assumption would be to demand at least first-order Sobolev regularity for control functions. The present paper is devoted to the study of an elliptic optimal control problem whose control function is chosen from a Sobolev space and has to satisfy additional equality constraints on its weak gradient. Noting that the associated Karush–Kuhn–Tucker conditions do not provide a necessary optimality condition for the underlying optimal control problem in general, one cannot simply solve the problem of interest by considering the system of first-order optimality conditions. Instead a penalization procedure with strong convergence properties for the computational solution is suggested and its computational implementation is studied in detail. Particularly, some essential difficulties arising from the gradient constraints which do not appear in standard optimal control are discussed.

Keywords:

• Control gradient constraints
• enforcement phenomena in FEM
• optimal control
• optimality conditions
• penalty method

AMS Subject Classifications:

• 35F05
• 49K20
• 49M25
• 90C48

Acknowledgements

The authors appreciate discussions with Gerd Wachsmuth which led to the discovery of an error in a previous version of the manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Yu Deng http://orcid.org/0000-0001-6303-3000

Patrick Mehlitz http://orcid.org/0000-0002-9355-850X

Uwe Prüfert http://orcid.org/0000-0001-5150-9008

Additional information

Funding

This work is partially supported by the DFG grant Analysis and Solution Methods for Bilevel Optimal Control Problems within the Priority Program SPP 1962 Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization [grant number DE 650/10-1].