The asymptotics of constrained control in optimal singular parabolic problems

A complete asymptotic solution is constructed and justified for the optimal singular parabolic problems with constrained control and a completely degenerate differential part of an operator.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 10, pp. 1345–1355, October, 1993.     

Global Attractor of One Nonlinear Parabolic Equation

We apply the theory of multivalued semiflows to a nonlinear parabolic equation of the reaction–diffusion type in the case where it is impossible to prove the uniqueness of its solution. A multivalued semiflow is generated by solutions satisfying a certain estimate global in time. We establish the existence of a global compact attractor in the phase space for the multivalued semiflow generated by a nonlinear parabolic equation. We prove that this attractor is an upper-semicontinuous function of a parameter.     

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.     

Attractors of Differential Inclusions and Their Approximation

We investigate the properties of solutions of differential inclusions in a Banach space. We prove a theorem on the existence of a global attractor for a multivalued semidynamical system generated by these solutions and a theorem on the approximation of an attractor in the Hausdorff metric.     

Qualitative Behavior of Solutions of a Randomly Perturbed Reaction-Diffusion Equation

On the basis of the developed abstract theory of random attractors of probability dissipative systems, we investigate the qualitative behavior of solutions of a nonuniquely solvable reaction-diffusion equation perturbed by a stochastic “cadlag” process. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 2, pp. 174–185, April–June, 2005.     

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.     

On the solvability of one class of problems of optimal control by coefficients in the principal part of a nonlinear elliptic operator

We consider the problem of the existence of solutions of an optimal-control problem for a nonlinear elliptic equation with Dirichlet conditions on the boundary in the case where the control functions are the coefficients in the principal part of the differential operator. It is shown that this problem has an optimal solutions in the class of generalized solenoidal matrices. Translated from Neliniini Kolyvannya, Vol. 12, No. 1, pp. 59–72, January–March, 2009.     

Approximate bounded synthesis for one weakly nonlinear boundary-value problem

We construct and substantiate an approximate control in the form of feedback for the problem of approximate bounded synthesis with semidefinite quality criterion for a parabolic equation containing a nonlinear term that depends regularly on a small parameter.