Moment exponential stability analysis of Markovian jump stochastic differential equations with uncertain transition jump rates

This paper is concerned with the moment exponential stability analysis of Markovian jump stochastic differential equations. The equations under consideration are more general, whose transition jump rates matrix Q is not precisely known. Sufficient conditions for testing the stability of such equations are established, and some numerical examples to illustrate the effectiveness of our results are presented.     

Exponential stability of semi-Markovian jump generalized neural networks with interval time-varying delays

This draft addresses the exponential stability problem for semi-Markovian jump generalized neural networks (S-MJGNNs) with interval time-varying delays. The exponential stability conditions are derived by establishing a suitable Lyapunov–Krasovskii functional and applying new analysis method. Improved results are obtained to guarantee the exponential stability of S-MJGNNs through improved reciprocally convex combination and new weighted integral inequality techniques. The method in this paper shows the advantages over some existing ones. To verify the advantages and benefits of employing proposed method is explained through numerical examples.

     

p-Moment stability of stochastic differential equations with impulsive jump and Markovian switching

This paper introduces some new concepts of p-moment stability for stochastic differential equations with impulsive jump and Markovian switching. Some stability criteria of p-moment stability for stochastic differential equations with impulsive jump and Markovian switching are obtained by using Liapunov function method. An example is also discussed to illustrate the efficiency of the obtained results.     

Stochastic stability properties of jump linear systems

Jump linear systems are defined as a family of linear systems with randomly jumping parameters (usually governed by a Markov jump process) and are used to model systems subject to failures or changes in structure. The authors study stochastic stability properties in jump linear systems and the relationship among various moment and sample path stability properties. It is shown that all second moment stability properties are equivalent and are sufficient for almost sure sample path stability, and a testable necessary and sufficient condition for second moment stability is derived. The Lyapunov exponent method for the study of almost sure sample stability is discussed, and a theorem which characterizes the Lyapunov exponents of jump linear systems is presented. Finally, for one-dimensional jump linear system, it is proved that the region for δ-moment stability is monotonically converging to the region for almost sure stability at δ↓0⁺     

Security control of positive semi-Markovian jump systems with actuator faults

Actuator faults usually cause security problem in practice.This paper is concerned with the security control of positive semi-Markovian jump systems with actuat...     

Stochastic stabilization of hybrid differential equations

This paper aims to determine whether or not a stochastic state feedback control can stabilize a given linear or nonlinear hybrid system. New methods are developed and sufficient conditions on the stability for hybrid stochastic differential equations are provided. These results are then used to examine stochastic stabilization by stochastic feedback controls. Two types of structure controls, namely state feedback and output injection, are discussed. Our stabilization criteria are in terms of linear matrix inequalities (LMIs) whence the feedback controls can be designed more easily in practice. A couple of examples and computer simulations are worked out to illustrate our theory.     

Stochastic stabilisation of functional differential equations

In this paper we investigate the problem of stochastic stabilisation for a general nonlinear functional differential equation. Given an unstable functional differential equation dx(t)/dt=f(t,x_t), we stochastically perturb it into a stochastic functional differential equation , where Σ is a matrix and B(t) a Brownian motion while X_t={X(t+θ):-τθ0}. Under the condition that f satisfies the local Lipschitz condition and obeys the one-side linear bound, we show that if the time lag τ is sufficiently small, there are many matrices Σ for which the stochastic functional differential equation is almost surely exponentially stable while the corresponding functional differential equation dx(t)/dt=f(t,x_t) may be unstable.     

Stochastic stability of a class of unbounded delay neutral stochastic differential equations with general decay rate

Without the linear growth condition on the drift coefficient, this article examines the existence and uniqueness of global solutions of a class of neutral stochastic differential equations with unbounded delay and their asymptotic stabilities with general decay rate. To illustrate the application of our results, this article gives a two-dimensional system as an example.     

Stochastic stability analysis of integral non-homogeneous Markov jump systems

This paper investigates the problem of stability analysis for time-delay integral Markov jump systems with time-varying transition rates. Some free-weight matrices are addressed and sufficient conditions are established under which the system is stochastically stable. The bound of delay is larger than those in other results obtained, which guarantees that the proposed conditions are tighter. Numerical examples show the effectiveness of the method proposed.     

Set stability of differential equations with time delay

Set stability and uniform set stability involve known specific bounds on the solutions of differential equations. These concepts are extended to include the case of differential equations with time delay. Liapunov-like theorems are presented which yield sufficient conditions for the set stability of such systems. Examples are given to demonstrate the method.     

Comments on "Stochastic stability of jump linear systems"

For original paper see ibid., vol.47, p.1204-8 (2002). The purpose of the article is to show that the conjecture presented in Remark 3 of the above paper by Y. Fang and K. Loparo (2002) is actually true.     

Stochastic differential equations for linear smoothing problems

Stochastic differential equations for the linear fixed point, fixed interval, and fixed lag smoothing problems are derived using the martingale representation theory.     

One-step methods for retarded differential equations with parameters

In this paper we develop a class of one-step methods combined with iterative methods to approximate the solution of retarded differential equations with parameters. The convergence theorems are established including the estimation of errors. Some methods are illustrated by three numerical examples.     

Non-fragile reliable control of nonlinear positive semi-Markovian jump systems with nonlinear actuator faults

This paper investigates the non-fragile reliable control of nonlinear positive semi-Markovian jump systems with nonlinear actuator faults. First, a novel fault model consisting of linear and nonlinear terms is established for the systems. By constructing a stochastic co-positive Lyapunov function, the non-fragile reliable controller is designed for the systems with additive gain perturbations using matrix decomposition technique. Then, the proposed design is extended for dealing with multiplicative gain perturbations and variable gain perturbations. Under the designed controllers, the resulting closed-loop systems are positive and stochastically stable. All presented conditions are solvable in terms of linear programming. Finally, two illustrative examples are provided to verify the effectiveness of the theoretical results.     

Stability of diffusion stochastic functional differential equations with Markov parameters

The paper justifies the second Lyapunov method for diffusion stochastic functional differential equations with Markov parameters, which generalize stochastic diffusion equations without aftereffect. Analogs of Lyapunov stability theorems, which generalize the results for systems with finite aftereffect, are proved. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 74–88, January–February 2008.     

The stability of second-order quadratic differential equations

This paper investigates the stability properties of second-order systems.dot{x} = f(x), wheref(x)contains either quadratic terms-system (1)-or linear and quadratic terms-system (2)-inx. The principal contributions are summarized in two theorems which give necessary, and sufficient conditions for stability and asymptotic stability in the large of systems (1) and (2), respectively.