Abstract

A measurable sweeping process with a multivalued perturbation is considered in a separable Hilbert space. The retraction of the sweeping process is a function of bounded variation whose differential measure is absolutely continuous with respect to a positive Radon measure. The values of the multivalued perturbation are convex compact sets. By a solution of the sweeping process we mean a right continuous function of bounded variation whose differential measure is absolutely continuous with respect to some positive Radon measure and the density of the differential measure with respect to the Radon measure satisfies the corresponding differential inclusion. In order to study the existence of solutions and properties of the solution set, we introduce the topology of uniform convergence on the space of right continuous functions of bounded variation and we investigate the compactness of sets in this space. Under the most general assumptions similar to those used to deal with absolutely continuous solutions, we prove the existence of solutions and the compactness of the solution set.

Author information

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

Alexander A. Tolstonogov

Matrosov Institute for System Dynamics and Control Theory, Russian Academy of Sciences, Irkutsk 664033, Russia

aatol@icc.ru

Keywords

  • Function of bounded variation
  • Radon measure
  • differential measure
  • density of a measure
  • BV solutions
  • existence theorem

Mathematics Subject Classification

28B20, 49J53

References

  1. N. Bourbaki: Integration, Chapitre V, Hermann, Paris (1967).Google Scholar
  2. N. Bourbaki: General Topology. Bacis Structures, Nauka, Moscow (1968).Google Scholar
  3. H. Brezis: Opérateurs Maximaux Monotones et Semi-Groupes de Contractious dans les Espaces de Hilbert, North-Holland, Amsterdam (1973).Google Scholar
  4. N. Dinculeanu: Vector Measures, VEB Deutscher Verlag der Wissenschaften, Berlin (1966).Google Scholar
  5. J. F. Edmond, L. Thibault: BV solutions of nonconvex sweeping process differential inclusion with perturbation, J. Differ. Equations 226 (2006) 135–179.https://doi.org/10.1016/j.jde.2005.12.005Google Scholar
  6. C. J. Himmelberg: Measurable relations, Fundamenta Math. 87 (1975) 53–72.https://doi.org/10.4064/fm-87-1-53-72Google Scholar
  7. F. Ky: Fixed point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci. USA 38 (1952) 121–126.https://doi.org/10.1073/pnas.38.2.121Google Scholar
  8. P. Mattila: Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press, Cambridge (1995).https://doi.org/10.1017/CBO9780511623813Google Scholar
  9. M. D. P. Monteiro Marques: Differential Inclusions in Nonsmooth Mechanical Problems: Shocks and Dry Friction, Birkhäuser, Basel (1993).Google Scholar
  10. J. J. Moreau: Sur les mesures différentielles des fonctions vectorielles à variation bornée, Sem. Anal. Convexe Montpellier, Exposé 17 (1975).Google Scholar
  11. J. J. Moreau: Evolution problem associated with a moving convex set in a Hilbert space, J. Diff. Equations 26 (1977) 347–374.https://doi.org/10.1016/0022-0396(77)90085-7Google Scholar
  12. J. J. Moreau, M. Valadier: A chain rule involving vector functions of bounded variation, J. Funct. Analysis 74 (1987) 333–345.https://doi.org/10.1016/0022-1236(87)90029-2Google Scholar
  13. F. Nacry: Perturbed BV sweeping process involving prox-regular sets, J. Nonlinear Convex Analysis 18 (2017) 1619–1651.Google Scholar
  14. L. Schwartz: Analyse Mathématique. Cours Professé à l’École Polytechnique, Paris, Hermann (1967).Google Scholar
  15. L. Thibault: Moreau sweeping process with bounded truncated retraction, J. Convex Analysis 23(4) (2016) 1051–1098.https://doi.org/10.68381/jca23039Google Scholar
  16. A. A. Tolstonogov: Properties of the space of proper functions, Math. Notes, Academy of Sciences of the USSR 35(4) (1984) 422–427.https://doi.org/10.1007/BF01139944Google Scholar
  17. A. A. Tolstonogov: Existence and relaxation of solutions for a subdifferential inclusion with unbounded perturbation, J. Math. Anal. Appl. 447 (2017) 269–288.https://doi.org/10.1016/j.jmaa.2016.09.061Google Scholar

Suggested citation

A. A. Tolstonogov. “Compactness of BV Solutions of a Convex Sweeping Process of Measurable Differential Inclusion.” Journal of Convex Analysis 27 (2020), No. 2, 673–695. https://doi.org/10.68381/jca27034

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Published by Heldermann Verlag, 2020. Rights now held by Banach Press LLC.