Stability for nonlinear delay differential equations of unstable type under impulsive perturbations

In the present paper, we obtain sufficient conditions for stability of the zero solution of the nonlinear delay differential equation

under impulsive perturbations, and show that the stability is caused by impulses, where τ > 0, f ∈ C([t₀, ∞) × R, R).     

Stability of autoresonance models subject to random perturbations for systems of nonlinear oscillation equations

Systems of differential equations arising in the theory of nonlinear oscillations in resonance-related problems are considered. Of special interest are solutions whose amplitude increases without bound with time. Specifically, such solutions correspond to autoresonance. The stability of autoresonance solutions with respect to random perturbations is analyzed. The classes of admissible perturbations are described. The results rely on information on Lyapunov functions for the unperturbed equations.     

Spectral convergence and nonlinear dynamics of reaction-diffusion equations under perturbations of the domain

In this paper we obtain the continuity of attractors for semilinear parabolic problems with Neumann boundary conditions relatively to perturbations of the domain. We show that, if the perturbations on the domain are such that the convergence of eigenvalues and eigenfunctions of the Neumann Laplacian is granted then, we obtain the upper semicontinuity of the attractors. If, moreover, every equilibrium of the unperturbed problem is hyperbolic we also obtain the continuity of attractors. We also give necessary and sufficient conditions for the spectral convergence of Neumann problems under perturbations of the domain.     

Stability of nonlinear stochastic-evolution equations

Criteria for boundedness, asymptotic stability of sample paths given by solutions to nonlinear stochastic-evolution equations are presented. The analysis is based on a functional Itô formula, Liapunov and related functionals, and generalization of methods developed in finite dimensions. Applications to parabolic Itô equations are given.     

Stability of solutions of stochastic wave equations with the Bessel operator under Poisson perturbations

We obtain sufficient conditions for the stability of solutions of deterministic and the corresponding stochastic wave equations with the Bessel operator under Poisson perturbations. The sufficient conditions are expressed in terms of coefficients of the equations, which allows us to construct domains of stability in the parameter space.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 7, pp. 974–978, July, 1990.     

Multivalued perturbations for a class of nonlinear evolution equations

We study in this paper the initial value problem for the multivalued differential equation wheref is in MS(2) andG is a multifunction fromC([0, T];) into the closed subsets of L²(0, Y;), satisfying suitable regularity assumptions. As an application we prove a local existence result for the problem

Stability region of nonlinear integrodifferential equations

This paper is devoted to the investigation of the asymptotic or exponential stability region for a class of nonlinear integrodifferential equations. We obtain two theorems to determine the stability region using the properties of nonnegative matrices and the techniques of inequalities. The main theorems are illustrated by two examples.     

Stability of solutions of nonlinear systems with unbounded perturbations

We study systems of differential equations with perturbations that are unbounded functions of time. We suggest a method for constructing Lyapunov functions to determine conditions under which the perturbations do not affect the asymptotic stability of the solutions. Translated fromMatematicheskie Zametki, Vol. 63, No. 1, pp. 3–8, January, 1998.     

Stability of solutions of nonlinear systems with unbounded perturbations

We study systems of differential equations with perturbations that are unbounded functions of time. We suggest a method for constructing Lyapunov functions to determine conditions under which the perturbations do not affect the asymptotic stability of the solutions.     

非线性脉冲差分方程解的稳定性

应用不等式估值法讨论了非线性脉冲时滞差分方程解的性质,并得到它的解的一致稳定性的一些充分条件.     

On perturbations of abstract fractional differential equations by nonlinear operators

We prove the unique solvability of a Cauchy-type problem for an abstract parabolic equation containing fractional derivatives and a nonlinear perturbation term. The result is applied to establish the solvability of the inverse coefficient problem for a fractional-order equation.     

Higher-dimensional nonlinear integrable equations: Asymptotics of solutions and perturbations

The survey is devoted to the study of solutions of (2 + 1)-dimensional (two spatial variables and time) integrable equations decreasing in spatial directions. As main representatives of these equations, the author considers the Kadomtsev-Petviashvili, Davey-Stewartson, and Ishimori equations.Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 5, Asymptotic Methods, 2003.     

Synthesis of nonlinear control systems invariant under perturbations

For nonlinear control systems with perturbations, we consider the problem of synthesis of perturbation-invariant characteristics (invariant functions) with the use of feedbacks. The existence of invariant functions is related to a decomposition of the control system for which the quotient system is independent of perturbations. We present conditions for the existence of such quotient systems, which are certain systems of partial differential equations. The synthesizing controls are found from these equations.     

Stability of some nonlinear functional differential equations

The procedure to construct Liapunov functionals for some nonlinear functional differential equations (FDEs) is proposed. Stability conditions for some nonlinear FDEs are obtained.Work partially supported by the Italian M.U.R.S.T. National Project Problemi non lineari ...