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
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.
Mosco convergence, nonconvex integrands, optimal control.
PDF file (752 kB)
Sbornik: Mathematics, 2011, 202:4, 583–619
MSC: 49J53, 49K40
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
\paper Variational stability of optimal control problems involving subdifferential operators
\jour Mat. Sb.
\jour Sb. Math.
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This publication is cited in the following articles:
Tolstonogov A.A., “Investigation of a new class of control systems”, Dokl. Math., 85:2 (2012), 178–180
S. A. Timoshin, “Variational stability of some optimal control problems describing hysteresis effects”, SIAM J. Control Optim., 52:4 (2014), 2348–2370
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
A. A. Tolstonogov, “Subdifferential inclusions with unbounded perturbation: existence and relaxation theorems”, Dokl. Math., 94:1 (2016), 396–400
A. A. Tolstonogov, “Control sweeping processes”, J. Convex Anal., 23:4 (2016), 1099–1123
Tolstonogov A.A., “Existence and relaxation of solutions for a subdifferential inclusion with unbounded perturbation”, J. Math. Anal. Appl., 447:1 (2017), 269–288
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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