状态约束最优控制中的非退化反常、能控性和间隙现象
Nondegenerate abnormality, controllability, and gap phenomena in optimal control with state constraints
摘要：
在最优控制理论中，下确界间隙是指在给定的最小值问题和通过扩大原始解和控制集而得到的扩展问题的下确界值之间存在间隙。空隙现象对于原始控制系统的可控性问题对于扩展解决方案来说有些“对偶”。在本文中，我们提出了充分条件的情况下，下确界差距和可控性的广泛的一类最优控制问题的端点和状态约束。这些条件是基于非退化版本的非光滑约束最大值原理，表示在次微分。特别地，在一些新的约束规格条件下，我们证明了：（i）如果一个扩展极小元是一个非退化的正规极值，则没有间隙出现;（ii）给定一个证明了约束条件的扩展解，它要么是一个非退化的异常极值，要么原系统对它是可控的。一个应用于自由终时的脉冲扩展，一个具有控制多项式动力学的非凸优化问题说明了结果。
英文摘要：
In optimal control theory, infimum gap means that there is a gap between the infimum values of a given minimum problem and an extended problem, obtained by enlarging the set of original solutions and controls. The gap phenomenon is somewhat "dual" to the problem of the controllability of the original control system to an extended solution. In this paper we present sufficient conditions for the absence of an infimum gap and for controllability for a wide class of optimal control problems subject to endpoint and state constraints. These conditions are based on a nondegenerate version of the nonsmooth constrained maximum principle, expressed in terms of subdifferentials. In particular, under some new constraint qualification conditions, we prove that: (i) if an extended minimizer is a nondegenerate normal extremal, then no gap shows up; (ii) given an extended solution verifying the constraints, either it is a nondegenerate abnormal extremal, or the original system is controllable to it. An application to the impulsive extension of a free end-time, non-convex optimization problem with control-polynomial dynamics illustrates the results.
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