Siberian Mathematical Journal

, Volume 44, Issue 4, pp 695–712 | Cite as

Approximation of Attainable Sets of an Evolution Inclusion of Subdifferential Type

  • A. A. Tolstonogov


In a separable Hilbert space we consider an evolution inclusion with a multivalued perturbation and evolution operators that are subdifferentials of a proper convex lower semicontinuous function depending on time. Along with the original inclusion, we consider a sequence of approximating evolution inclusions with the same perturbation and the evolution operators that are subdifferentials of the Moreau–Yosida regularizations of the original function. We show that the attainable set of the original inclusion, regarded as a multivalued function of time, is the uniform (in time) limit in the Hausdorff metric of the sequence of attainable sets of the approximating inclusions. As an application we consider an example of a control system with discontinuous nonlinearity.

subdifferential Moreau–Yosida regularization continuous selection extreme point attainable set discontinuous nonlinearity 


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  1. 1.
    Brezis H., Operateurs Maximaux Monotones, North-Holland, Amsterdam (1973).Google Scholar
  2. 2.
    Bourbaki N., Topological Vector Spaces [Russian translation], Izdat. Inostr. Lit., Moscow (1959).Google Scholar
  3. 3.
    Kenmochi N., “Solvability of nonlinear evolution equations with time-dependent constraints and applications,” Bull. Fac. Educ. Chiba Univ., 30, 1–87 (1981).Google Scholar
  4. 4.
    Himmelberg C. J., “Measurable relations,” Fund. Math., 87, No. 1, 53–72 (1975).Google Scholar
  5. 5.
    Kenmochi N., “On the quasi-linear heat equation with time-dependent obstacles,” Nonlinear Anal. Theory Methods Appl., 5, No. 1, 71–80 (1981).Google Scholar
  6. 6.
    Bourbaki N., General Topology [Russian translation], Nauka, Moscow (1975).Google Scholar
  7. 7.
    Tolstonogov A. A. and Tolstonogov D. A., “L p-continuous extreme selectors of multifunctions with decomposable values: Existence theorems,” Set-Valued Anal., 4, 173–203 (1996).Google Scholar
  8. 8.
    Fryszkowski A., “Continuous selections for a class of nonconvex multivalued maps,” Studia Math., 76, No. 2, 163–174 (1983).Google Scholar
  9. 9.
    Tolstonogov A. A. and Tolstonogov D. A., “L p-continuous extreme selectors of multifunction with decomposable values: Relaxation theorems,” Set-Valued Anal., 4, 237–269 (1996).Google Scholar
  10. 10.
    Tolstonogov A. A., “L p-Continuous selections of fixed points of multifunctions with decomposable values. I: Existence theorems,” Sibirsk. Mat. Zh., 40, No. 3, 695–709 (1999).Google Scholar
  11. 11.
    Ekeland I. and Temam R., Convex Analysis and Variational Problems [Russian translation], Mir, Moscow (1979).Google Scholar
  12. 12.
    Cebuhar W. A., “Approximate linearization of control systems with discontinuous non-linearities,” Optimal Control Appl. Methods, 16, 341–359 (1995).Google Scholar
  13. 13.
    Filippov A. F., Differential Equations with a Discontinuous Right Side [in Russian], Nauka, Moscow (1985).Google Scholar

Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • A. A. Tolstonogov
    • 1
  1. 1.Institute of System Dynamics and Control TheoryIrkutsk

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