In this paper, we study optimal control problems for stochastic semilinear partial differential equations, which lack the maximum principle, and whose coefficients do not have bounded Frechet derivatives. We propose an approximation scheme for the corresponding optimization problem, and prove convergence of the approximating solutions on both finite and infinite time intervals.

We study stability and robustness of global attractors for parabolic inclusions subjected to external disturbances entering through the boundary conditions. To this end we develop an approach consisting in investigation of a suitable family of non-autonomous semiprocesses possessing a uniform attractor, which depends on disturbances. Based on the property of its upper semicontinuity, we establish conditions guaranteeing the asymptotic gain property for the global attractor of the unperturbed system w.r.t. disturbances.

In this work we consider invariant measures for neutral type stochastic delay evolution equations in Hilbert space. We derive the conditions for the existence and uniqueness of invariant measure for neutral type stochastic delay evolution equation. The limiting behavior of invariant measures is also investigated in the asymptotic limit, as the length of delay interval tends to zero.

We consider two interconnected systems, each being input-to-state (ISS) stable with respect to a corresponding set Ai, i = 1,2, and look for stability conditions guaranteeing that the whole interconnection is ISS with respect to some set A. We explain that the relation between the given A1,A2 and unknown A to be found is nontrivial and develop a small-gain theorem for this kind of interconnections, which allows to derive a suitable set A. The issue of minimality of A remains open. Motivating example, which also demonstrates the applicability of our result is provided.

We consider the stochastic thin-film equation with linear deterministic and stochastic Itô perturbations. The existence of nonnegative weak martingale solutions on the semi-axis is established, and their asymptotic behavior as t →∞ is investigated. It is shown that in square mean the L^∞ norm of the solution converges to the spatial mean value of the initial condition, multiplied by a random factor similar to a geometric Wiener process.

We consider a parabolic inclusion with an upper semicontinuous multivalued interaction function satisfying the sign and growth conditions of the reaction-diffusion type. We prove the global solvability of the corresponding initial-boundary-value problem in the phase space L^2 and establish the existence of a global attractor. The conditions for boundedness of the attractor in the space L^∞ are obtained.

In this paper, we investigate the qualitative behavior of an evolutionary problem that consists of a semilinear parabolic equation whose trajectories undergo instantaneous impulsive perturbations at the moments when some integral functional reaches a certain threshold value. The key object is the uniform attractor of the corresponding impulsive infinite-dimensional dynamical system. The novelty of this study is the analysis of mild solutions in the phase space of continuous functions. Under general assumptions on the impulsive parameters, we prove that this problem generates an impulsive dynamical system, and its trajectories have a compact uniform attractor with respect to the supremum norm (sup-norm).

The paper investigates the qualitative behavior of weak solutions to a dissipative infinite-dimensional non-autonomous perturbed system, which consists of a parabolic reaction-diffusion system and a nonlinear system of ordinary differential equations. Perturbations are modeled by bounded functions included in the right-hand side of both systems. The considered system is known to possess a global attractor in the absence of perturbations, which determines the long-term dynamics of all trajectories. However, the robustness of such attractors under external disturbances remains a challenging issue, especially in the context of nonlinear and infinite-dimensional systems. In this study, we adopt an approach based on the theory of uniform attractors for non-autonomous dynamical systems (semi-processes). A corresponding family of semi-processes associated with the perturbed PDE–ODE system is constructed. The existence of a uniform attractor is proven, and its convergence to the global attractor of the unperturbed system is established as the amplitude of disturbances tends to zero. This allows us to derive a nonlocal robust estimate of the asymptotic gain (AG) type, which complements the local ISS-based robustness result and provides an upper bound on the deviation of perturbed trajectories from the unperturbed attractor in terms of the perturbation magnitude. The results contribute to the theoretical foundation for robustness analysis in dissipative infinite-dimensional systems with complex long-term dynamics.

In this paper we consider stochastic thin-film equation with nonlinear drift terms, colored Gaussian Stratonovych noise, as well as nonlinear colored Wiener noise. By means of Trotter-Kato-type decomposition into deterministic and stochastic parts, we couple both of these dynamics via a discrete-in-time scheme, and establish its convergence to a non-negative weak martingale solution.

In this work we establish existence and uniqueness of weak solutions of nonlinear stochastic functional differential equations of neutral type in Hilbert spaces. We also study the continuous dependence of their solutions on the initial data. Our approach is based on Galerkin approximation scheme.

In this paper, we study the stochastic reaction-diffusion equations in the case when the reaction coefficient does not satisfy the traditional sign and growth conditions. We establish existence of a weak solution and investigate the long-time behavior in terms of the existence of invariant measure.

This article is dedication to the life and scientific path of the famous Ukrainian mathematician M.O. Perestyuk. The problems of the differential equations with impulse action theory, which was one of the leading M.O. Perestyuk’s research directions, are illustrated by examples accessible to a wide audience.

The paper deals with the impulsive infinite-dimensional problem generated by solutions of the thermal conductivity equation under the condition of impulse ”pumping” of heat. Moments of impulses are not fixed and are determined by the amount of total heat in the system. It is proved that such a problem generates impulsive dynamical system in the space of continuous functions and its ω-limits sets are investigated.

This paper presents machine learning methods for approximate solutions of reaction-diffusion equations with multivalued interaction functions. This approach addresses the challenge of finding all possible solutions for such equations, which often lack uniqueness. The proposed method utilizes physics-informed neural networks (PINNs) to approximate generalized solutions.

In this paper we investigate the optimal control problem for a parabolic differential inclusion with rapidly oscillating variables in the finite interval. There are many approaches intended for the investigation of control problems for differential equations and inclusions. Thus, in particular, the asymptotic methods are used fairly extensively. Among these methods, we can especially mention the averaging method, which was mathematically rigorously substantiated by Krylov M.M. and Bogolyubov M.M. The well-known Krasnoselski–Krein theorem and its multi-valued analogue play an essential role for the investigation of the above-mentioned problems. The averaging method was substantiated, in particular, for ordinary differential inclusions, inclusions with partial derivatives, and inclusions with the Hukuhara derivative. When dealing with multi-valued mappings one faces specific problems, such as closedness, convexity of the family of solutions, existence of limit solutions, selection of solutions with given properties, etc. However, the well-developed apparatus of mathematical analysis applied to the study of multi-valued functions makes it possible to apply the averaging method to the optimal control problem described above. Thus, using the averaging method the convergence of optimal controls and optimal trajectories of solutions of the exact problem to optimal control and the trajectory of the averaged problem is proved in the paper.

We consider an optimal control problem for a differential inclusion of the Carathéodory type affine with respect to the control with a coercive cost functional on a semiaxis and with fast oscillating time-dependent coefficients. We prove that, when the small parameter converges to zero, the solution to this problem tends to some solution of the optimal control problem with averaged coefficients, where the averaging we understand in the sense of the Kuratowski upper limit.

In this paper we establish the existence of the uniform attractor for a semi linear parabolic problem with bounded non autonomous disturbances in the phase space of continuous functions. We applied obtained results to prove the asymptotic gain property with respect to the global attractor of the undisturbed system.

In this paper, we establish the existence of global attractor for a semilinear parabolic problem with non-Lipschitz nonlinear term in the phase space of continuous functions.