SIAM Journal on Control and Optimization, Volume 58, Issue 4, Page 2678-2709, January 2020.
This paper deals with optimal control problems described by a controlled version of Moreau's sweeping process governed by convex polyhedra, where measurable control actions enter additive perturbations. This class of problems, which addresses unbounded discontinuous differential inclusions with intrinsic state constraints, is truly challenging and underinvestigated in control theory while being highly important for various applications. To attack such problems with constrained measurable controls, we develop a refined method of discrete approximations with establishing its well-posedness and strong convergence. This approach, married to advanced tools of first-order and second-order variational analysis and generalized differentiations, allows us to derive adequate collections of necessary optimality conditions for local minimizers, first in discrete-time problems and then in the original continuous-time controlled sweeping process by passing to the limit. The new results include an appropriate maximum condition and significantly extend the previous ones obtained under essentially more restrictive assumptions. We compare them with other versions of the maximum principle for controlled sweeping processes that have been recently established for global minimizers in problems with smooth sweeping sets by using different techniques. The obtained necessary optimality conditions are illustrated by several examples.

中文翻译：

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约束不连续控制对扰动清扫过程的优化

SIAM控制与优化杂志，第58卷，第4期，第2678-2709页，2020年1月。
本文讨论了由凸多面体控制的Moreau扫描过程的受控版本描述的最佳控制问题，其中可测量的控制动作会导致加扰。这类问题解决了具有固有状态约束的无穷不连续微分包含问题，在控制理论中确实具有挑战性和研究不足，同时对于各种应用也非常重要。为了用可测量的约束控制这些问题，我们开发了一种精确的离散近似方法，确立了它的适定性和强收敛性。这种方法与一阶和二阶变分分析和广义微分的高级工具相结合，使我们能够为局部极小化子得出足够的必要最优性条件集合，首先在离散时间问题中，然后在原始连续时间控制的扫描过程中达到极限。新的结果包括一个适当的最大条件，并大大扩展了在实质上更具限制性的假设下获得的先前条件。我们将它们与最大原则的其他版本进行比较，该原则是最近通过使用不同技术为全局最小化程序建立的具有平滑清扫集问题的全局最小化方法。所获得的必要最优条件通过几个例子来说明。我们将它们与控制扫描过程的最大原理的其他版本进行比较，该方法最近已通过使用不同技术针对全局最小化器在平滑扫描集问题中建立了全局最小化器。所获得的必要最优条件通过几个例子来说明。我们将它们与最大原则的其他版本进行比较，该原则是最近通过使用不同技术为全局最小化程序建立的具有平滑清扫集问题的全局最小化方法。所获得的必要最优条件通过几个例子来说明。