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

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.

DOI: https://doi.org/10.4213/sm7704

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

Bibliographic databases:

UDC: 517.977.57
MSC: 49J53, 49K40

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
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• http://mi.mathnet.ru/eng/msb7704
• https://doi.org/10.4213/sm7704
• http://mi.mathnet.ru/eng/msb/v202/i4/p123

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
2. S. A. Timoshin, “Variational stability of some optimal control problems describing hysteresis effects”, SIAM J. Control Optim., 52:4 (2014), 2348–2370
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
4. A. A. Tolstonogov, “Subdifferential inclusions with unbounded perturbation: existence and relaxation theorems”, Dokl. Math., 94:1 (2016), 396–400
5. A. A. Tolstonogov, “Control sweeping processes”, J. Convex Anal., 23:4 (2016), 1099–1123
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
7. A. A. Tolstonogov, “Bogolyubov's theorem for a controlled system related to a variational inequality”, Izv. Math., 84:6 (2020), 1192–1223
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