Journal of Convex Analysis 23 (2016), No. 4, 1099--1123
Copyright Heldermann Verlag 2016
Control Sweeping Processes
Alexander A. Tolstonogov
Institute for System Dynamics and Control Theory, Siberian Branch of the Russian Academy of Sciences, Lermontov str. 134, Irkutsk 664033, Russia
A control system with internal and external controls is considered in a finite dimensional space. The internal control is a multivalued function of time with convex closed possibly unbounded values, while the external control is represented by a measurable function of time. The internal control acts on the system by normal cones at the points of control values entering the right-hand side of the system. The external control enters the right-hand side of the system as a perturbation. The admissible internal controls are taken from a relatively compact set in the corresponding metric space, whereas the constraint for the external control is a multivalued mapping with closed nonconvex values. Along with the original system we consider an extended system in which the internal control is taken from the closure of the set of admissible controls of the original system, and the external control constraint is the convexified constraint of the original system. Minimization problems for integral functionals over the solutions of the original and extended systems with a nonconvex and the convexified in the exterior control variable integrand are considered. The integrand depends on the phase variable, the internal and external controls. Existence of solutions and relationships between solution sets of the original and extended systems and solutions of optimization problems are investigated.
Keywords: Sweeping process, evolution inclusion, perturbation, normal cone, subdifferential.
MSC: 46E35; 32U20, 31C15
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