Strong Solutions of Stochastic Differential Inclusions with Unbounded Right-Hand Side in a Hilbert Space

In a separable Hilbert space, a stochastic differential inclusion with coefficients whose values are closed not necessarily convex sets is considered. Two existence theorems for strong solutions are proved. In the first theorem, the proof is based on the use of Euler polygonal lines; in the second, on the successive approximation method. Instead of the assumption that the coefficients of the inclusion are globally Lipschitz, which is traditional in such cases, some conditions that are less restrictive for the problems in question are used.     

An Approximation Scheme for Reflected Stochastic Differential Equations with Non-Lipschitzian Coefficients

Journal of Theoretical Probability - In this paper, we study a numerical approximation scheme for reflected stochastic differential equations (SDEs) with non-Lipschitzian coefficients in a bounded...     

Discrete Approximation of Impulsive Differential Inclusions

The article deals with the approximation of the solution set and the reachable sets of an impulsive differential inclusion with variable times of impulses. It is strongly connected to [11 T. Donchev ( 2008 ). Approximation of the solution set of impulsive systems . In Lecture Notes in Comput. Sci . ( et al. . eds.). Vol. 4818 , Springer-Verlag , Berlin , pp. 309 – 316 . [Google Scholar]] and is its continuation. We achieve order of convergence 1 for the Euler approximation under Lipschitz assumptions on the set-valued right-hand side and on the functions describing the jump surfaces and jumps themselves. Another criterion prevents the beating phenomena and generalizes available conditions. Several test examples illustrate the conditions and the practical evaluation of the jump conditions.     

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.     

Internal Layers for a Singularly Perturbed Second-Order Quasilinear Differential Equation with Discontinuous Right-Hand Side

A singularly perturbed boundary value problem for a second-order quasilinear ordinary differential equation is studied. We consider a new class of problems in which the nonlinearities experience discontinuities, which leads to the appearance of sharp transition layers in a neighborhood of the points of discontinuity. The existence of solutions is proved, and their asymptotic expansion with an internal transition layer is constructed.     

Stability Theorems in the First-Order Approximation for Differential Inclusions

We establish multi-valued and infinite-dimensional versions of stability theorems in the first-order approximation. The differential inclusions treated as first-order approximations can be nonautonomous and, in several cases under study, nonhomogeneous with respect to the phase variable. We outline applications in stability theory of solutions to parabolic inclusions.     

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

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.     

Weak Exponential Stability for Time-Periodic Differential Inclusions via First Approximation Averaging

In this work we propose a method to study a weak exponential stability for time-varying differential inclusions applying an averaging procedure to a first approximation. Namely, we show that a weak exponential stability of the averaged first approximation to the differential inclusion implies the weak exponential stability of the original time-varying inclusion. The result is illustrated by an example.     

On Systems of Boundary Value Problems for Differential Inclusions

Herein we consider the existence of solutions to second-order, two-point boundary value problems （BVPs） for systems of ordinary differential inclusions. Some new Bernstein-Nagumo conditions are presented that ensure a priori bounds on the derivative of solutions to the differential inclusion. These a priori bound results are then applied, in conjunction with appropriate topological methods, to prove some new existence theorems for solutions to systems of BVPs for differential inclusions. The new conditions allow of the treatment of systems of BVPs without growth restrictions.     

带奇异系数的有限维与无穷维随机微分方程

本文对具非Lipschitz系数的随机微分方程给出解的存在唯一性与非爆炸性的新判别条件,少许改进了文\cite{4}的有关结果. 通过控制交互作用, 该结果还被推广到无穷维情形.     

Feedback in a Control Problem for a System with Discontinuous Right-Hand Side

Differential Equations - On a finite time horizon, we consider a control system described by a vector differential equation with right-hand side that changes its structure at some times spaced by a...     

Properties of a Solution of an Inhomogeneous Hyperbolic Equation with Random Right-Hand Side

We consider an inhomogeneous hyperbolic equation with zero initial and boundary conditions and a random centered sample-continuous Gaussian right-hand side. We establish conditions for the existence of a solution of the first boundary-value problem of mathematical physics in the form of a series uniformly convergent in probability in terms of a covariance function. An estimate for the distribution of the supremum of a solution of this problem is obtained. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 4, pp. 474–482, April, 2005.     

Morse Decompositions for Periodic General Dynamical Systems and Differential Inclusions

We first establish the Morse decomposition theory of periodic invariant sets for non-autonomous periodic general dynamical systems (set-valued dynamical systems). Then we discuss the stability of Morse decompositions of periodic uniform forward attractors. We also apply the abstract results to non-autonomous periodic differential inclusions with only upper semi-continuous right-hand side. We show that Morse decompositions are robust with respect to both internal and external perturbations (upper semi-continuity of Morse sets). Finally as an application we study the effect of small time delays to asymptotic behavior of control systems from the point of view of Morse decompositions.