具有非正则混合约束的控制问题的非退化条件
Nondegeneracy Conditions for Control Problems with Nonregular Mixed Constraints
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中文总结 AI 辅助
该研究针对带非正则混合约束的最优控制问题，建立含纯有限可加集函数（电荷）的非退化与正规性条件，提出四个可验证条件并通过实例验证其可实现性与局限。
中文摘要 AI 辅助
我们针对带有单个非正则混合不等式约束的最优控制问题建立了非退化与正规性条件，其中纯有限可加集函数（电荷）作为乘子出现。首先，我们针对在单个时刻非正则的模型问题证明，退化是内在且不可避免的：所有容许过程都通过纯电荷以退化方式满足必要条件，而自然正则近似的归一化乘子在$L^\flat([0,1];\bbR)^*$中不存在弱$^*$极限，仅在巴拿赫极限意义下的每个广义极限下得到纯电荷。受此分析启发，我们提出四个可验证条件，逐步保证非退化乘子、纯电荷消失、正规性以及电荷质量的明确界。此外，我们证明在非正则终端时刻，存在平衡恒等式将电荷质量与成本乘子关联，决定完全非正规性与正规性之间的选择。我们提供实例确认各条件的可实现性并讨论其局限。
英文摘要
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$^*$ limit in $L^\infty([0,1];\mathbb{R})^*$, 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.
