Polyhedral multivalued mappings: properties and applications

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.