Pullback attractors for a class of extremal solutions of the 3D Navier-Stokes system

In this paper we construct a dynamical process (in general, multivalued) generated by the set of solutions of an optimal control problem for the three-dimensional Navier-Stokes system. We prove the existence of a pullback attractor for such multivalued process. Also, we establish the existence of a uniform global attractor containing the pullback attractor. Moreover, under the unproved assumption that strong globally defined solutions of the three-dimensional Navier-Stokes system exist, which guaranties the existence of a global attractor for the corresponding multivalued semiflow, we show that the pullback attractor of the process coincides with the global attractor of the semiflow.     

Optimal control over the heat field in thin bodies local constraints on control

Asymptotic solutions of problems of optimal locally constrained control over the heat field in thin bodies are constructed and justified. These problems relate to the critical case of singularly perturbed systems (the degenerate problem has a family of solutions). Dnepropetrovsk Technical University of Railway Transport, Dnepropetrovsk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 48, No. 9, pp. 1200–1208, September, 1996.     

Global Attractor for a Nonautonomous Inclusion with Discontinuous Right-Hand Side

We consider a nonautonomous inclusion the upper and lower selectors of whose right-hand side are determined by functions with discontinuities of the first kind. We prove that this problem generates a family of multivalued semiprocesses for which there exists a global attractor compact in the phase space.     

Asymptotics of locally constrained control in optimal parabolic problems

We construct and justify asymptotic solutions of optimal parabolic problems with locally constrained control which depends only on time.     

On positive solutions of one class of evolutionary inclusions of the subdifferential type

We obtain sufficient conditions for the existence of nonnegative solutions of the evolutionary inclusions of subdifferential type with multivalued non-Lipschitz perturbations and, under the additional condition of dissipativity, prove the existence of a global attractor in the class of nonnegative functions.     

Asymptotic behaviour of reaction–diffusion equations with non-damped impulsive effects

In this paper we consider non-autonomous reaction–diffusion systems with impulsive effects at fixed moments of time from the point of view of the theory of global attractors. For a translation-compact nonlinear term which does not provide the uniqueness of the Cauchy problem, and for different classes of non-damped multivalued impulse perturbations, we construct a multivalued non-autonomous dynamical system and prove for it the existence of a compact global attractor.