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Mat. Sb., 2011, Volume 202, Number 4, Pages 123–160 (Mi msb7704)  

This article is cited in 7 scientific papers (total in 7 papers)

Variational stability of optimal control problems involving subdifferential operators

A. A. Tolstonogov

Institute of System Dynamics and Control Theory, Siberian Branch of the Russian Academy of Sciences

Abstract: This paper is concerned with the problem of minimizing an integral functional with control-nonconvex integrand over the class of solutions of a control system in a Hilbert space subject to a control constraint given by a phase-dependent multivalued map with closed nonconvex values. The integrand, the subdifferential operators, the perturbation term, the initial conditions and the control constraint all depend on a parameter. Along with this problem, the paper considers the problem of minimizing an integral functional with control-convexified integrand over the class of solutions of the original system, but now subject to a convexified control constraint. By a solution of a control system we mean a ‘trajectory-control’ pair. For each value of the parameter, the convexified problem is shown to have a solution, which is the limit of a minimizing sequence of the original problem, and the minimal value of the functional with the convexified integrand is a continuous function of the parameter. This property is commonly referred to as the variational stability of a minimization problem. An example of a control parabolic system with hysteresis and diffusion effects is considered.
Bibliography: 24 titles.

Keywords: Mosco convergence, nonconvex integrands, optimal control.


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English version:
Sbornik: Mathematics, 2011, 202:4, 583–619

Bibliographic databases:

UDC: 517.977.57
MSC: 49J53, 49K40
Received: 01.03.2010

Citation: A. A. Tolstonogov, “Variational stability of optimal control problems involving subdifferential operators”, Mat. Sb., 202:4 (2011), 123–160; Sb. Math., 202:4 (2011), 583–619

Citation in format AMSBIB
\by A.~A.~Tolstonogov
\paper Variational stability of optimal control problems involving subdifferential operators
\jour Mat. Sb.
\yr 2011
\vol 202
\issue 4
\pages 123--160
\jour Sb. Math.
\yr 2011
\vol 202
\issue 4
\pages 583--619

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    This publication is cited in the following articles:
    1. Tolstonogov A.A., “Investigation of a new class of control systems”, Dokl. Math., 85:2 (2012), 178–180  crossref  mathscinet  zmath  isi  elib  elib  scopus
    2. S. A. Timoshin, “Variational stability of some optimal control problems describing hysteresis effects”, SIAM J. Control Optim., 52:4 (2014), 2348–2370  crossref  mathscinet  zmath  isi  elib  scopus
    3. N. I. Pogodaev, A. A. Tolstonogov, “The variational stability of an optimal control problem for Volterra-type equations”, Siberian Math. J., 55:4 (2014), 667–686  mathnet  crossref  mathscinet  isi
    4. A. A. Tolstonogov, “Subdifferential inclusions with unbounded perturbation: existence and relaxation theorems”, Dokl. Math., 94:1 (2016), 396–400  crossref  crossref  mathscinet  zmath  isi  elib  scopus
    5. A. A. Tolstonogov, “Control sweeping processes”, J. Convex Anal., 23:4 (2016), 1099–1123  mathscinet  zmath  isi
    6. Tolstonogov A.A., “Existence and relaxation of solutions for a subdifferential inclusion with unbounded perturbation”, J. Math. Anal. Appl., 447:1 (2017), 269–288  crossref  mathscinet  zmath  isi  scopus
    7. A. A. Tolstonogov, “Bogolyubov's theorem for a controlled system related to a variational inequality”, Izv. Math., 84:6 (2020), 1192–1223  mathnet  crossref  crossref
  • Математический сборник Sbornik: Mathematics (from 1967)
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