Stabilizability of a class of stochastic bilinear hybrid systems

The problem of the stabilizability of stochastic nonlinear hybrid systems with a Markovian or any switching rule is considered. Using the Lyapunov technique sufficient conditions for the asymptotic stabilizability in probability by a smooth controller in every structure are found. In particular, the asymptotic stabilizability in probability problem of stochastic bilinear hybrid systems with a Markovian or any switching rule is discussed and a closed-loop controller is found. Also the sufficient conditions for the exponential mean-square stabilizability for bilinear hybrid systems with any switching based on the Lie algebra approach are formulated and an open-loop controller is designed. The obtained results are illustrated by examples and simulations.     

Global Stability of an Autonomous Stochastic Delay Differential Equation with Discontinuous Coefficients

We study the stability and asymptotic stability of the zero solution of autonomous stochastic delay differential equations with discontinuous coefficients by the Lyapunov second method. For the equations under study, we obtain analogs of the Lyapunov stability theorem and the Barbashin–Krasovskii and Zubov asymptotic stability theorems for the zero solution.     

Stability Criteria for Solutions of Systems of Linear Deterministic or Stochastic Delay Difference Equations with Continuous Time

We give spectral and algebraic coefficient criteria (necessary and sufficient conditions) as well as sufficient algebraic coefficient conditions for the Lyapunov asymptotic stability of solutions to systems of linear deterministic or stochastic delay difference equations with continuous time under white noise coefficient perturbations for the case in which all delay ratios are rational. For stochastic systems, mean-square asymptotic stability is studied. The Lyapunov function method is used. Our criteria on algebraic coefficients and our sufficient conditions are stated in terms of matrix Lyapunov equations (for deterministic systems) and matrix Sylvester equations (for stochastic systems).     

Stabilizability of linear switching systems

A complete characterization of stabilizability for linear switching systems is not available in the literature. In this paper, we show that the asymptotic stabilizability of linear switching systems is equivalent to the existence of a hybrid Lyapunov function for the controlled system, for a suitable control strategy. Further, we prove that asymptotic stabilizability of a switching system with minimum dwell time, is equivalent to Input to State Stability (ISS) of the controlled switching system, with a stabilizing control law. We then derive some structural reductions of the hybrid state space, which allow a decomposition of the original problem into simpler subproblems. The relationships between this approach and the well-known Kalman decomposition of linear dynamic control systems are explored.     

STABILITY OF SOLUTIONS FOR CERTAIN THIRD-ORDER NONLINEAR STOCHASTIC DELAY DIFFERENTIAL EQUATIONS

In this paper, we investigate stochastic asymptotic stability of the zero solution for certain third-order nonlinear stochastic delay differential equations by constructing Lyapunov functionals.     

Some Contributions to Stochastic Asymptotic Stability and Boundedness via Multiple Lyapunov Functions

Most of the existing results on stochastic stability use a single Lyapunov function, but we shall instead use multiple Lyapunov functions in this paper to establish some sufficient criteria for locating the limit sets of solutions of stochastic differential equations. From them follow many useful results on stochastic asymptotic stability and boundedness, which enable us to construct the Lyapunov functions much more easily in applications. In particular, the well-known classical theorem on stochastic asymptotic stability is a special case of our more general results. These show clearly the power of our new results.     

Stability analysis of stochastic differential equations with the use of Lyapunov functions of constant sign

We prove a theorem on the asymptotic stability of stochastic differential equations using Lyapunov functions of constant sign.     

Attraction and Stochastic Asymptotic Stability and Boundedness of Stochastic Functional Differential Equations with Respect to Semimartingales

Abstract

In this paper, we will establish new results on the attraction for solutions to stochastic functional differential equations with respect to semimartingale. Most of the existing results stochastic stability use a single Lyapunov function, but we shall instead use multiple Lyapunov functions in the study of attraction. Moreover, from our results on the attraction follow several new criteria on almost surely asymptotic stability and boundedness of the solutions.     

Asymptotic Stability and Boundedness of Stochastic Differential Equations with Respect to Semimartingales

In this paper, we shall use multiple Lyapunov functions to establish some sufficient criteria for locating the limit sets of solutions of stochastic differential equations with respect to semimartingales. From them follow many useful results on stochastic asymptotic stability and boundedness, including some classical results as special cases. In particular, our new asymptotic stability criteria do not require the diffusion operator associated with the underlying stochastic differential equation be negative definite, while most of the existing results do require this negative definite property essentially.     

Equations for second moments of solutions of a system of linear differential equations with random semi-Markov coefficients and random input

We derive equations that determine second moments of a random solution of a system of Itô linear differential equations with coefficients depending on a finite-valued random semi-Markov process. We obtain necessary and sufficient conditions for the asymptotic stability of solutions in the mean square with the use of moment equations and Lyapunov stochastic functions.

     

Stochastic Nonlinear Beam Equations with Lévy Jump

In this paper, we study stochastic nonlinear beam equations with Lévy jump, and use Lyapunov functions to prove existence of global mild solutions and asymptotic stability of the zero solution.     

Localization of limit sets and stability of systems of the neutral type with nonmonotone Lyapunov functionals

We suggest new approaches to the study of the asymptotic stability of equilibria for equations of the neutral type. Nonmonotone indefinite Lyapunov functionals are used. We investigate the localization of solutions with respect to the level surfaces of a Lyapunov functional and a functional estimating the derivative of the Lyapunov functional along the solutions. By using solution localization tests, we obtain new conditions for the asymptotic stability of equilibria for equations of the neutral type with bounded right-hand side. We present asymptotic stability tests that do not impose any a priori stability condition on the difference operator. A generalization of the Barbashin–Krasovskii theorem for nonmonotone indefinite Lyapunov functionals is proved for autonomous equations.     

Criteria of the mean-square asymptotic stability of solutions of systems of linear stochastic difference equations with continuous time and delay

We obtain spectral and algebraic coefficient criteria and sufficient conditions for the mean-square asymptotic stability of solutions of systems of linear stochastic difference equations with continuous time and delay. We consider the case of a rational correlation between delays and a “white-noise”-type stochastic perturbation of coefficients. We use the method of Lyapunov functions. Most results are presented in terms of the Sylvester and Lyapunov matrix algebraic equations. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 8, pp. 1073–1081, August, 1998. This work was partially supported by the Joint Foundation of the Ukrainian Government and the Soros International Science Foundation (grant No. K42100).     

Stabilization of Passive Nonlinear Stochastic Differential Systems by Bounded Feedback

Abstract

The purpose of this paper is to give a systematic method for global asymptotic stabilization in probability of nonlinear control stochastic differential systems the unforced dynamics of which are Lyapunov stable in probability. The approach developed in this paper is based on the concept of passivity for nonaffine stochastic differential systems together with the theory of Lyapunov stability in probability for stochastic differential equations. In particular, we prove that, as in the case of affine in the control stochastic differential systems, a nonlinear stochastic differential system is asymptotically stabilizable in probability provided its unforced dynamics are Lyapunov stable in probability and some rank conditions involving the affine part of the system coefficients are satisfied. Furthermore, for such systems, we show how a stabilizing smooth state feedback law can be designed explicitly. As an application of our analysis, we construct a dynamic state feedback compensator for a class of nonaffine stochastic differential systems.     

随机脉冲微分方程的渐近p稳定性

本文研究的是随机脉冲微分方程的渐近p稳定性.首先给出一些预备知识,然后运用Lyapunov函数建立随机脉冲微分方程平凡解的渐近p稳定性的充分条件.     

Asymptotic autonomous attractors for a stochastic lattice model with random viscosity

We study forward asymptotic autonomy of a pullback random attractor for a non-autonomous random lattice system and establish the criteria in terms of convergence, recurrence, forward-pullback absorption and asymptotic smallness of the discrete random dynamical system. By applying the abstract result to both non-autonomous and autonomous stochastic lattice equations with random viscosity, we show the existence of both pullback and global random attractors such that the time-component of the pullback attractor semi-converges to the global attractor as the time-parameter tends to infinity.     

On practical asymptotic stabilizability of switched affine systems

In this paper, we report some new results on practical asymptotic stabilizability of switched systems consisting of affine subsystems. We first briefly review some practical asymptotic stabilizability notions and some results from our previous papers. Then we propose a new approach to estimate the region of attraction for switched affine systems. Based on this new approach, we present several new sufficient conditions for the practical asymptotic stabilizability and global practical asymptotic stabilizability of such systems. Finally, a computational approach to check the new sufficient conditions is proposed and it is applied to several numerical examples.     

Stability of Riccati's Equation with Random Stationary Coefficients

The purpose of this paper is to study under weak conditions of stabilizability and detectability, the asymptotic behavior of the matrix Riccati equation which arises in stochastic control and filtering with random stationary coefficients. We prove the existence of a stationary solution of this Riccati equation. This solution is attracting, in the sense that if P _t is another solution, then onverges to 0 exponentially fast as t tends to +∞ , at a rate given by the smallest positive Lyapunov exponent of the associated Hamiltonian matrices. Accepted 13 January 1998     

Stability analysis of stochastic functional differential equations with infinite delay and its application to recurrent neural networks

In this paper, we investigate the stochastic functional differential equations with infinite delay. Some sufficient conditions are derived to ensure the pth moment exponential stability and pth moment global asymptotic stability of stochastic functional differential equations with infinite delay by using Razumikhin method and Lyapunov functions. Based on the obtained results, we further study the pth moment exponential stability of stochastic recurrent neural networks with unbounded distributed delays. The result extends and improves the earlier publications. Two examples are given to illustrate the applicability of the obtained results.     

NORMAL切换系统在干扰下的性质

考虑了具有有界干扰NORMAL切换系统的稳定性分析,运用Lyapunov函数给出了干扰的界限.主要解决了切换系统在有界干扰下的稳定性分析,给出了当系统干扰满足一定的条件时,切换系统在任意的切换条件下轨迹渐近稳定.同时讨论了带干扰的线性切换系统的镇定问题.问题的讨论中对系统的要求少,因此更具一般性.