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Variational stability of optimal control problems involving subdifferential operators. (English. Russian original) Zbl 1220.49012

Sb. Math. 202, No. 4, 583-619 (2011); translation from Mat. Sb. 2011, No. 4, 123-160 (2011).
Summary: 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 are all parameter dependent. 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.

MSC:

49J53 Set-valued and variational analysis
49K40 Sensitivity, stability, well-posedness
49J45 Methods involving semicontinuity and convergence; relaxation
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