2018
Том 70
№ 7

All Issues

Kapustyan O. V.

Articles: 13
Article (Ukrainian)

Stability of global attractors of impulsive infinite-dimensional systems

Kapustyan O. V., Perestyuk N. A., Romanyuk I. V.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 1. - pp. 29-39

The stability of global attractor is proved for an impulsive infinite-dimensional dynamical system. The obtained abstract results are applied to a weakly nonlinear parabolic equation whose solutions are subjected to impulsive perturbations at the times of intersection with a certain surface of the phase space.

Article (Russian)

Global attractors of impulsive infinite-dimensional systems

Kapustyan O. V., Perestyuk N. A.

↓ Abstract

Ukr. Mat. Zh. - 2016. - 68, № 4. - pp. 517-528

We study the existence of global attractors in discontinuous infinite-dimensional dynamical systems, which may have trajectories with infinitely many impulsive perturbations. We also select a class of impulsive systems for which the existence of a global attractor is proved for weakly nonlinear parabolic equations.

Article (Ukrainian)

Approximate Synthesis of Distributed Bounded Control for a Parabolic Problem with Rapidly Oscillating Coefficients

Kapustyan O. V., Rusina A. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2015. - 67, № 3. - pp. 355-365

We study the problem of finding the optimal control in the form of feedback (synthesis) for a linear-quadratic problem in the form of a parabolic equation with rapidly oscillating coefficients and distributed control on the right-hand side (whose Fourier coefficients obey certain restrictions in the form of inequalities) and a quadratic quality criterion. We deduce the exact formula of synthesis and justify its approximate form corresponding to the replacement of rapidly oscillating coefficients by their averaged values.

Article (Ukrainian)

Approximate stabilization for a nonlinear parabolic boundary-value problem

Kapustyan O. A., Kapustyan O. V., Sukretna A. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2011. - 63, № 5. - pp. 654-661

For a problem of optimal stabilization of solutions of a nonlinear parabolic boundary-value problem with small parameter of a nonlinear summand, we justify the form of approximate regulator on the basis of the formula of optimal synthesis of the corresponding linear quadratic problem.

Article (Ukrainian)

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

Kapustyan O. V., Shklyar T. B.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2011. - 63, № 4. - pp. 472-480

Sufficient conditions of the existence of a nonnegative solution are obtained for an evolution inclusion of subdifferential type with multivalued non-Lipschitz perturbation. Under the additional condition of dissipativity, the existence of the global attractor in the class of nonnegative functions is proved.

Brief Communications (Ukrainian)

On the existence of global attractors for one class of cascade systems

Kapustyan O. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2004. - 56, № 9. - pp. 1287-1291

We investigate the qualitative behavior of solutions of cascade systems without uniqueness. We prove that solutions of a reaction-diffusion system perturbed by a system of ordinary differential equations and solutions of a system of equations of a viscous incompressible liquid with passive components form families of many-valued semiprocesses for which a compact global attractor exists in the phase space.

Article (Ukrainian)

Random attractors for ambiguously solvable systems dissipative with respect to probability

Kapustyan O. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2004. - 56, № 7. - pp. 892–900

We prove a theorem on the existence of a random attractor for a multivalued random dynamical system dissipative with respect to probability. Abstract results are used for the analysis of the qualitative behavior of solutions of a system of ordinary differential equations with continuous right-hand side perturbed by a stationary random process. In terms of the Lyapunov function, for an unperturbed system, we give sufficient conditions for the existence of a random attractor.

Article (Ukrainian)

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

Kapustyan O. V., Kasyanov P. O.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2003. - 55, № 11. - pp. 1467-1475

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.

Article (Russian)

Global Attractor of an Evolution Inclusion with Pulse Influence at Fixed Moments of Time

Kapustyan O. V., Perestyuk N. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2003. - 55, № 8. - pp. 1058-1068

We consider an autonomous evolution inclusion with pulse perturbations at fixed moments of time. Under the conditions of global solvability, we prove the existence of a minimal compact set in the phase space that attracts all trajectories.

Article (Ukrainian)

Averaged Synthesis of the Optimal Control for a Wave Equation

Kapustyan O. V., Sukretna A. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2003. - 55, № 5. - pp. 612-620

For a wave equation, we determine an optimal control in the feedback form and prove the convergence of the constructed approximate control to the exact one.

Article (Ukrainian)

Global Attractor of One Nonlinear Parabolic Equation

Kapustyan O. V., Shkundin D. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2003. - 55, № 4. - pp. 446-455

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.

Brief Communications (Ukrainian)

Attractors of Differential Inclusions and Their Approximation

Kapustyan O. V., Valero J.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2000. - 52, № 7. - pp. 975-979

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.

Brief Communications (Ukrainian)

An attractor of a semiflow generated by a system of phase-field equations without the uniqueness of a solution

Kapustyan O. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1999. - 51, № 7. - pp. 1006–1009

We prove the existence of a global compact attractor for a multivalued semiflow generated by a system of phase-field equations with conditions on nonlinearity that do not guarantee the uniqueness of a solution.