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

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, LICQ and MFCQ imply USC (for inequalities, the implication follows from Gordan's theorem). Unlike classical regularity conditions, which require differentiability of the active constraints, USC remains applicable when the terminal constraints are nondifferentiable, where LICQ and MFCQ are not defined; the converse implication does not hold in general. 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. The approach yields these classical results in a concise and transparent manner, avoiding the need for constructing cones of endpoint variations or applying separation theorems. From the computational perspective, the exact penalty formulation allows the differential constraints to be eliminated from the optimization entirely, so that only the cost functional and the terminal constraints are penalized. A numerical implementation for the time-optimal control of a harmonic oscillator confirms the practical applicability of the approach: the computed optimal time agrees with the analytical value obtained via Pontryagin's maximum principle to high accuracy. This establishes USC as a unifying regularity concept encompassing classical LICQ and MFCQ and extending naturally to nonsmooth settings.