Investigation of regularity conditions in optimal control problems with geometric mixed constraints

Abstract

We investigate regularity conditions in optimal control problems with mixed constraints of a general geometric type, in which a closed non-convex constraint set appears. A closely related question to this issue concerns the derivation of necessary optimality conditions under some regularity conditions on the constraints. By imposing strong and weak regularity condition on the constraints, we provide necessary optimality conditions in the form of Pontryagin maximum principle for the control problem with mixed constraints. The optimality conditions obtained here turn out to be more general than earlier results even in the case when the constraint set is convex. The proofs of our main results are based on a series of technical lemmas which are gathered in the Appendix.

Keywords:

• regularity conditions
• optimal control
• mixed constraints
• geometric constraints

AMS Subject Classifications:

• 49K15
• 49K21

Acknowledgements

The authors are very grateful to the anonymous referees for their help in improving the exposition of this paper.

Notes

No potential conflict of interest was reported by the author(s).

1 In the literature, the modulus of surjection is introduced for set-valued maps . If spaces and are finite dimensional, then

Here, is the limiting coderivative of at .[21] By definition, when . If we set , then .

2 As , the number tends to the number which is called modulus of regularity and designated as . It turns out that the modulus of regularity is the lower bound of all such for which the estimate of metric regularity still holds, see [21].

3 Here, we adopt the notation: for .

4 Here, and from now on, stands for the limiting subdifferential in the sense of Mordukhovich [21]. In finite dimensions, its convex hull coincides with the subdifferential in the sense of Clarke [27]. Thus, the function takes values in the Clarke’s subdifferential of the distance function.

Additional information

Funding

This work was supported by the Brazilian National Council for Scientific and Technological Development – CNPq (Brazil) [grant number 401689/2012-3] ‘Sem Fronteiras’, [grant number 309335/2012-4], [grant number 479109/2013-3]; the Russian Foundation for Basic Research [grant number 13-01-00494], [grant number 14-01-31185], [grant number 15-01-04601]; FCT [grant number PEst-OE-EEI-UI0147-2014], [grant number PTDC/EEI-AUT/1450/2012]; Sao Paulo State Foundation (FAPESP) [grant number 2013/07375-0] – through the Center for Mathematical Sciences Applied to Industry – CeMEAI/CEPID; the Ministry of Education and Science of the Russian Federation [grant number 1.333.2014/K].