Nina V. Zadoianchuk

Average Cost Markov Decision Processes with Weakly Continuous Transition Probabilities

This paper presents sufficient conditions for the existence of stationary optimal policies for average cost Markov decision processes with Borel state and action sets and weakly continuous transition probabilities. The one-step cost functions may be unbounded, and the action sets may be noncompact. The main contributions of this paper are: (i) general sufficient conditions for the existence of stationary discount optimal and average cost optimal policies and descriptions of properties of value functions and sets of optimal actions, (ii) a sufficient condition for the average cost optimality of a stationary policy in the form of optimality inequalities, and (iii) approximations of average cost optimal actions by discount optimal actions.

On Global Attractors of Multivalued Semiprocesses and Nonautonomous Evolution Inclusions

In the first chapter, we considered the existence and properties of global attractors for autonomous multivalued dynamical systems. When the equation is nonautonomous, new and challenging difficulties appear. In this case, if uniqueness of the Cauchy problem holds, then the usual semigroup of operators becomes a two-parameter semigroup or process [38, 39], as we have to take into account the initial and the final time of the solutions.

Fatou's Lemma for Weakly Converging Probabilities

Fatous lemma states under appropriate conditions that the integral of the lower limit of a sequence of functions is not greater than the lower limit of the integrals. This note describes similar inequalities when, instead of a single measure, the functions are integrated with respect to different measures that form a weakly convergent sequence.

Long-time behavior of solutions for quasilinear hyperbolic hemivariational inequalities with application to piezoelectricity problem

Abstract We consider quasilinear autonomous inclusions of hyperbolic type. The dynamics of all weak solutions defined on the positive semi-axis of time is studied. We prove the existence of trajectory and global attractors and investigate their structure. The classes of mathematical models for piezoelectric fields containing the multidimensional law are studied. The conditions of the output of each weak solution for this problem at stationary states are given. We consider as a particular case the piezoelectric model for PZT-4 piezoceramics as one of the possible applications.

Long-Time Behaviour of Solutions for Autonomous Evolution Hemivariational Inequality with Multidimensional “Reaction-Displacement” Law

We consider autonomous evolution inclusions and hemivariational inequalities with nonsmooth dependence between determinative parameters of a problem. The dynamics of all weak solutions defined on the positive semiaxis of time is studied. We prove the existence of trajectory and global attractors and investigate their structure. New properties of complete trajectories are justified. We study classes of mathematical models for geophysical processes and fields containing the multidimensional “reaction-displacement” law as one of possible application. The pointwise behavior of such problem solutions on attractor is described.

A criterion for the existence of strong solutions for the 3D Navier–Stokes equations

Abstract In this note we provide a criterion for the existence of globally defined solutions for any regular initial data for the 3D Navier–Stokes system in Serrin’s classes.

Topological Properties of Strong Solutions for the 3D Navier-Stokes Equations

In this chapter we give a criterion for the existence of global strong solutions for the 3D Navier-Stokes system for any regular initial data.

Properties of Resolving Operator for Nonautonomous Evolution Inclusions: Pullback Attractors

One of the most effective approaches to investigate nonlinear problems, represented by partial differential equations, inclusions and inequalities with boundary values, consists in the reduction of them into differential-operator inclusions, in infinite-dimensional spaces governed by nonlinear operators. In order to study these objects, the modern methods of nonlinear analysis have been used [7, 10, 11, 26]. Convergence of approximate solutions to an exact solution of the differential-operator equation or inclusion is frequently proved on the basis of the property of monotony or pseudomonotony of the corresponding operator.

Abstract Theory of Multivalued Semiflows

Beginning from the pioneering works [3, 52], the theory of global attractors of infinite-dimensional dynamical systems has become one of the main objects for investigation. Since then, deep results about existence, properties, structure, and dimension of global attractors for a wide class of dissipative systems have been obtained (see, e.g., [7, 38, 54, 75, 78]). For the application of this classical theory to partial and functional differential equations, it was necessary to have global existence and uniqueness of solutions of the Cauchy problem for all initial data of the phase space.

Attractors for Lattice Dynamical Systems

A lot of processes coming from Physics, Chemistry, Biology, Economy, and other sciences can be described using systems of reaction-diffusion equations. In this chapter, we study the asymptotic behavior of the solutions of a system of infinite ordinary differential equations (a lattice dynamical system) obtained after the spacial discretization of a system of reaction-diffusion equations in an unbounded domain. This kind of dynamical systems is then of importance in the numerical approximations of physical problems.