Kapustyan O. V.
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.
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.
Approximate Synthesis of Distributed Bounded Control for a Parabolic Problem with Rapidly Oscillating Coefficients
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.
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.
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.
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.
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.
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.
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.
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.
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.