Optimality Conditions for a Constrained Control Problem
SIAM Journal on Control and Optimization
Nonsmooth analysis and control theory
Nonsmooth analysis and control theory
Weak maximum principle for optimal control problems with mixed constraints
Nonlinear Analysis: Theory, Methods & Applications
Optimal Control Problems with Set-Valued Control and State Constraints
SIAM Journal on Optimization
Optimality Conditions: Abnormal and Degenerate Problems (Mathematics and its Applications Volume 526)
Necessary Conditions for Constrained Problems under Mangasarian-Fromowitz Conditions
SIAM Journal on Control and Optimization
Optimal Control Problems with Mixed Constraints
SIAM Journal on Control and Optimization
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The search for multiplier rules in dynamic optimization has been an important theme in the subject for over a century; it was central in the classical calculus of variations, and the Pontryagin maximum principle of optimal control theory is part of this quest. A more recent thread has involved problems with so-called mixed constraints involving the control and state variables jointly, a subject which now boasts a considerable literature. Recently, Clarke and de Pinho proved a general multiplier rule for such problems that extends and subsumes rather directly most of the available results, namely, those which postulate some kind of rank condition or, more generally, a constraint qualification (or generalized Mangasarian-Fromowitz condition). An exception to this approach is due to Schwarzkopf, whose well-known theorem replaces the rank hypothesis, for relaxed problems, by one of covering. The purpose of this article is to show how to obtain this type of theorem from the general multiplier rule of Clarke and de Pinho. In so doing, we subsume, extend, and correct the currently available versions of Schwarzkopf's result.