Necessary Conditions for Adverse Control Problems Expressed by Relaxed Derivatives

TL;DR

This paper introduces a novel framework for deriving necessary conditions in adverse control problems with two players, especially those with prior knowledge, using relaxed derivatives and regularization techniques.

Contribution

It develops a new set of necessary conditions expressed via relaxed derivatives for adverse control problems, extending classical results to more complex strategic scenarios.

Findings

Conditions expressed in terms of relaxed derivatives.

Dual variables are limits of computable sequences.

Applicable to minimax and robust control problems.

Abstract

This paper provides a framework for deriving a new set of necessary conditions for adverse control problems among two players. The distinguish feature of such problems is that the first player has a priori knowledge on the second player strategy. A subclass of adverse control problems is the one of minimax control problems, which frequently arise in robust dynamic optimization. The conditions derived in this manuscript are expressed in terms of relaxed derivatives: the dual variables and the related functions are limits of computable sequences, obtained by considering a regularized version of the original problem and applying well known necessary condition. This topic was initially treated by J. Warga.