A Unified Regularity Condition for Optimal Control: Bridging LICQ, MFCQ, and Subdifferentials

TL;DR

This paper introduces the Unified Separation Condition (USC), a regularity criterion that generalizes classical conditions for optimal control problems, applicable even with nondifferentiable constraints, and demonstrates its effectiveness through theoretical derivations and numerical examples.

Contribution

The paper proposes the USC, unifying classical regularity conditions for optimal control and extending applicability to nondifferentiable constraints, simplifying derivations of transversality conditions.

Findings

USC generalizes LICQ and MFCQ for nondifferentiable constraints.

Transversality conditions derived under USC are concise and classical in form.

Numerical example confirms the practical effectiveness of the approach.

Abstract

This paper presents a unified derivation of transversality conditions in optimal control problems using exact penalty functions. The key regularity condition is that the origin is uniformly separated from the subdifferential of the penalty function in a neighborhood of the admissible set. This condition, hereafter referred to as the Unified Separation Condition (USC), generalizes the classical Mangasarian-Fromovitz condition for inequalities and linear independence of gradients for equalities; in the smooth case, these classical conditions are equivalent to USC, as shown via Gordan's theorem. The USC remains applicable even when constraint functions are nondifferentiable, where classical constraint qualifications are not defined. Assuming exactness, we derive transversality conditions for all major cases: fixed and free terminal time, equality and inequality constraints, moving manifolds, and free left endpoint. Remarkably, this approach yields these classical results in a concise and transparent manner, avoiding the need for constructing cones of endpoint variations or applying separation theorems. The theoretical results are complemented by a numerical implementation applied to the time-optimal control of a harmonic oscillator. The numerical implementation converges to the exact solution obtained via Pontryagin's maximum principle combined with transversality conditions, confirming the consistency and practical applicability of the proposed methodology.