Duality in optimal control with first order partial differential inclusions

Abstract

For the first time in this paper, the dual problem is constructed for the problem with generalized first order partial differential inclusions, the duality theorem is proved. For discrete problems, necessary and sufficient optimality conditions are derived in the form of the Euler–Lagrange type inclusion. Thus, it is possible to construct dual problems for problems with partial differential inclusions on the basis of dual operations of addition and infimal convolution of convex functions. To pass from the dual problem to the discrete-approximation problem, important equivalence theorems are proved, without which it is unlikely that certain success can be achieved along this path. Hence, we believe that this method of constructing dual problems can serve as the only possible method for studying duality for a wide class of problems with partial/ordinary differential inclusions. The results obtained are demonstrated on some linear problem and on a problem with first-order polyhedral partial differential inclusions.

KEYWORDS:

• Approximate
• discrete and partial differential inclusions
• Euler–Lagrange
• first order
• polyhedral
• necessary and sufficient

AMS 2020 Mathematics Subject Classifications:

• 49K20
• 49N15
• 49M25

Acknowledgment

The author expresses his heartfelt gratitude to the anonymous reviewers whose comments and suggestions have improved the work.

Disclosure statement

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