On the practical global uniform asymptotic stability of stochastic differential equations

The method of Lyapunov functions is one of the most effective ones for the investigation of stability of dynamical systems, in particular, of stochastic differential systems. The main purpose of the paper is the analysis of the stability of stochastic differential equations by using Lyapunov functions when the origin is not necessarily an equilibrium point. The global uniform boundedness and the global practical uniform exponential stability of so- lutions of stochastic differential equations based on Lyapunov techniques are investigated. Furthermore, an example is given to illustrate the applicability of the main result.Comment: To appear in Stochastic

Growth conditions for the stability of a class of time-varying perturbed singular systems

summary:In this paper, we investigate the problem of stability of linear time-varying singular systems, which are transferable into a standard canonical form. Sufficient conditions on exponential stability and practical exponential stability of solutions of linear perturbed singular systems are obtained based on generalized Gronwall inequalities and Lyapunov techniques. Moreover, we study the problem of stability and stabilization for some classes of singular systems. Finally, we present a numerical example to validate the effectiveness of the abstract results of this paper

Lyapunov functionals and practical stability for stochastic differential delay equations with general decay rate

This paper stands for the almost sure practical stability of nonlinear stochastic differential delay equations (SDDEs) with a general decay rate. We establish some sufficient conditions based upon the construction of appropriate Lyapunov functionals. Furthermore, we provide some numerical examples to validate the effectiveness of the abstract results of this paper

Practical exponential stability in mean square of stochastic partial differential equations

The main aim of this paper is to establish some criteria for the mean square and almost sure practical exponential stability of a nonlinear monotone stochastic partial differential equations

Practical asymptotic stability of nonlinear stochastic evolution equations

In this paper we establish some su cient conditions ensuring almost sure practical asymptotic stability with a non-exponential decay rate for solutions to stochastic evolution equations based on Lyapunov techniques

Practical stability of stochastic delay evolution equations

In this paper we investigate the almost sure practical stability for a class of stochastic functional evolution equations. We establish some sufficient conditions based on the construction of appropriate Lyapunov functional. The abstract results are then applied to some illustrative examples.Fondo Europeo de Desarrollo Regional (FEDER)Ministerio de Economía y Competitividad (España)Junta de Andalucí

Practical exponential stability of impulsive stochastic functional differential equations

This paper is devoted to the investigation of the practical exponential stability of impulsive stochastic functional differential equations. The main tool used to prove the results is the Lyapunov-Razumikhin method which has proven very useful in dealing with stability problems for differential systems when the delays involved in the equations are not differentiable but only continuous. An illustrative example is also analyzed to show the applicability and interest of the main results.Fondo Europeo de Desarrollo RegionalMinisterio de Economía y CompetitividadJunta de Andalucí

Partial stability analysis of stochastic differential equations with a general decay rate

This paper is concerned with the almost sure partial practical stability of stochastic differential equations with general decay rate. We establish some sufficient conditions based upon the construction of appropriate Lyapunov functions. Finally, we provide a numerical example to demonstrate the efficiency of the obtained results

Estimates of exponential convergence for solutions of stochastic nonlinear systems

This paper aims to analyze the behavior of the solutions of a stochastic perturbed system with respect to the solutions of the stochastic unperturbed system. To prove our stability results, we have derived a new Gronwall–type inequality instead of the Lyapunov techniques, which makes it easy to apply in practice and it can be considered as a more general tool in some situations. On the one hand, we present sufficient conditions ensuring the global practical uniform exponential stability of SDEs based on Gronwall’s inequalities. On the other hand, we investigate the global practical uniform exponential stability with respect to a part of the variables of the stochastic perturbed system by using generalized Gronwall’s inequalities. It turns out that, the proposed approach gives a better result comparing with the use of a Lyapunov function. A numerical example is presented to illustrate the applicability of our results