We study the multivalued mappings on a closed interval of the real line whose values are polyhedra in a separable Hilbert space. The polyhedron space is endowed with the metric of the Mosco convergence of sequences of closed convex sets. A polyhedron is defined as the intersection of finitely many closed half-spaces. The equations of the corresponding hyperplanes involve normals and reals. The normals and reals for a polyhedral multivalued mapping depend on time and are regarded as internal controls. The space of polyhedral multivalued mappings is endowed with the topology of uniform convergence. We study the properties of sets in the space of polyhedral mappings expressed in terms of internal controls. Applying the results, we establish the existence of solutions to polyhedral sweeping processes and study the dependence of solutions on internal controls. We consider minimization problems for integral functionals over the solutions to controlled polyhedral sweeping processes which, along with internal controls, have traditional measurable controls called external.

中文翻译：

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多面体多值映射：属性和应用

我们研究了实线的闭区间上的多值映射，实线的值是可分希尔伯特空间中的多面体。多面体空间被赋予了闭凸集序列的 Mosco 收敛度量。多面体被定义为有限多个封闭半空间的交集。对应超平面的方程涉及法线和实数。多面体多值映射的法线和实数取决于时间并被视为内部控制。多面体多值映射空间具有一致收敛的拓扑结构。我们研究了用内部控制表示的多面体映射空间中集合的性质。应用结果，我们确定多面体扫描过程的解决方案的存在，并研究解决方案对内部控制的依赖性。我们考虑积分泛函的最小化问题，而不是受控多面体扫描过程的解决方案，该过程与内部控制一起具有称为外部的传统可测量控制。