Nondegeneracy Conditions for Control Problems with Nonregular Mixed Constraints

Abstract

We establish nondegeneracy and normality conditions for optimal control problems with a single nonregular mixed inequality constraint, where purely finitely additive set functions (charges) appear as multipliers. We first prove for a model problem nonregular at a single instant that degeneration is intrinsic and unavoidable: every admissible process satisfies the necessary conditions degenerately via a pure charge, and normalized multipliers of natural regular approximations admit no weak${}^{\ast }$ limit in $L^{\infty }([0,1];\mathbb{R}{)}^{\ast }$, yielding only pure charges under every generalized limit, in the sense of Banach limits. Motivated by this analysis, we propose four verifiable conditions that progressively guarantee nondegenerate multipliers, the vanishing of the pure charge, normality, and explicit bounds on charge mass. Furthermore, we show that at a nonregular terminal instant, a balance identity ties the charge mass to the cost multiplier, deciding between complete abnormality and normality. We provide examples confirming the realizability and discussing the limits of each condition.