Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

Alexander A. Tolstonogov

Institute for System Dynamics and Control Theory, Siberian Branch of the Russian Academy of Sciences, Lermontov str. 134, Irkutsk 664033, Russia

aatol@icc.ru

Keywords

  • Sweeping process
  • evolution inclusion
  • perturbation
  • normal cone
  • subdifferential

Mathematics Subject Classification

46E35; 32U20, 31C15

References

  1. H. Attouch: Variational Convergence for Functions and Operators, Pitman, Boston (1984).Google Scholar
  2. J. P. Aubin, A. Cellina: Differential Inclusions, Springer, Berlin (1984).https://doi.org/10.1007/978-3-642-69512-4Google Scholar
  3. G. Colombo, R. Henrion, N. D. Hoang, B. S. Mordukhovich: Discrete approximations of a controlled sweeping process, Set-Valued Var. Anal. 23 (2015) 69–86.https://doi.org/10.1007/s11228-014-0299-yGoogle Scholar
  4. G. Colombo, R. Henrion, N. D. Hoang, B. S. Mordukhovich: Optimal control of sweeping process, Dyn. Contin. Discrete Impuls. Syst., Ser. B, Appl. Algorithms 19 (2012) 117–159.Google Scholar
  5. I. Ekeland, R. Temam: Convex Analysis and Variational Problems, Elsevier, New York (1976).Google Scholar
  6. S. Gutman: Topological equivalence in the space of integrable vector-valued functions, Proc. Amer. Math. Soc. 93 (1985) 40–42.https://doi.org/10.1090/S0002-9939-1985-0766523-2Google Scholar
  7. C. J. Himmelberg: Measurable relations, Fundam. Math. 87 (1975) 53–72.https://doi.org/10.4064/fm-87-1-53-72Google Scholar
  8. K. Kuratowski: Topology. Vol. I, Academic Press, New York (1966).Google Scholar
  9. E. Michael: Continuous selections. I, Ann. Math. (2) 63 (1956) 361–382.https://doi.org/10.2307/1969615Google Scholar
  10. J. J. Moreau: Evolution problem associated with a moving convex set in Hilbert space, J. Differ. Equations 26 (1977) 347–374.https://doi.org/10.1016/0022-0396(77)90085-7Google Scholar
  11. A. A. Tolstonogov: Compactness in the space of set-valued mappings with closed values, Dokl. Math. 89(3) (2014) 293–295.https://doi.org/10.1134/S1064562414030120Google Scholar
  12. A. A. Tolstonogov: Continuity in the parameter of the minimum value of an integral functional over the solutions of an evolution control system, Nonlinear Anal. 75(12) (2012) 4711–4727.https://doi.org/10.1016/j.na.2011.12.029Google Scholar
  13. A. A. Tolstonogov: Investigation of a new class of control systems, Dokl. Math. 85(2) (2012) 178–180.https://doi.org/10.1134/S1064562412020056Google Scholar
  14. A. A. Tolstonogov: Sweeping process with unbounded nonconvex perturbation, Nonlinear Anal. 108 (2014) 291–301.https://doi.org/10.1016/j.na.2014.06.002Google Scholar
  15. A. A. Tolstonogov: Variational stability of optimal control problems involving subdifferential operators, Sb. Math. 202(4) (2011) 583–619.https://doi.org/10.1070/SM2011v202n04ABEH004157Google Scholar

Suggested citation

A. A. Tolstonogov. “Control Sweeping Processes.” Journal of Convex Analysis 23 (2016), No. 4, 1099–1123. https://doi.org/10.68381/jca23040

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2016. Rights now held by Banach Press LLC.