pth Moment exponential stability of impulsive stochastic functional differential equations and application to control problems of NNs

This paper investigates the pth moment exponential stability of impulsive stochastic functional differential equations. Some sufficient conditions are obtained to ensure the pth moment exponential stability of the equilibrium solution by the Razumikhin method and Lyapunov functions. Based on these results, we further discuss the pth moment exponential stability of generalized impulsive delay stochastic differential equations and stochastic Hopfield neural networks with multiple time-varying delays from the impulsive control point of view. The results derived in this paper improve and generalize some recent works reported in the literature. Moreover, we see that impulses do contribute to the stability of stochastic functional differential equations. Finally, two numerical examples are provided to demonstrate the efficiency of the results obtained.     

The pth moment boundedness of stochastic functional differential equations with Markovian switching

This paper gives some Razumikhin-type theorems on pth moment boundedness of stochastic functional differential equations with Markovian switching (SFDEwMS) by using Razumikhin technique and comparison principle. Some improved conditions on pth moment stability are also proposed. The main results of this paper allow the estimated upper bound of the diffusion operator associated with the underlying SFDEwMS of the Lyapunov function to have time-varying coefficients (the coefficients may even be sign-changing functions). Examples are provided to illustrate the effectiveness of the proposed results.     

Mean square exponential input-to-state stability of stochastic Markovian reaction-diffusion systems with impulsive perturbations

Mean square exponential input-to-state stability (MSEISS) is considered for stochastic Markovian reaction-diffusion systems (SMRDSs) with impulsive perturbations. Both the boundary input and distributed input are considered in SMRDSs. With the Lyapunov–Krasovskii functional method, impulse theory and inequality techniques, a sufficient condition is established to achieve the MSEISS for SMRDSs with completely known transition rate matrix. Moreover, combined with the obtained sufficient conditions, the effects of the impulse and diffusion terms on MSEISS are demonstrated by examples. Then, the case is studied that the transition rate matrix is partially unknown and sufficient conditions are presented to ensure the MSEISS in light of the introduced free constants. Finally, two numerical examples are given to illustrate the validity of our theoretical results.     

Quantitative mean square exponential stability and stabilization of stochastic systems with Markovian switching

This paper is concerned with the quantitative mean square exponential stability and stabilization for stochastic systems with Markovian switching. First, the concept of quantitative mean square exponential stability(QMSES) is introduced, and two stability criteria are derived. Then, based on an auxiliary definition of general finite-time mean square stability(GFTMSS), the relations among QMSES, GFTMSS and finite time stochastic stability (FTSS) are obtained. Subsequently, QMSE-stabilization is investigated and several new sufficient conditions for the existence of the state and observer-based controllers are provided by means of linear matrix inequalities. An algorithm is given to achieve the relation between the minimum states’ upper bound and the states’ decay velocity. Finally, a numerical example is utilized to show the merit of the proposed results.     

On moment exponential stability of Markovian switching integral delay systems

This paper studies the moment exponential stability analysis of a class of Markovian switching integral delay systems (MSIDSs). The existence, uniqueness and stability of the solution are discussed firstly. Secondly, by selecting appropriate Lyapunov-Krasovskii (L-K) functionals, delay-dependent sufficient conditions are given such that the general form of MSIDSs and the special form of MSIDSs having multiple delays are mean square exponentially stable respectively. The results are then generalized to robust stability of MSIDSs having multiple delays with uncertain parameters. Finally, numerical examples are given to illustrate the effectiveness of the proposed theoretical results.     

Robust exponential stability of uncertain impulsive stochastic neural networks with delayed impulses

In this paper, by using Lyapunov functions, Razumikhin techniques and stochastic analysis approaches, the robust exponential stability of a class of uncertain impulsive stochastic neural networks with delayed impulses is investigated. The obtained results show that delayed impulses can make contribution to the stability of system. Compared with existing results on related problems, this work improves and complements ones from some works. Two examples are discussed to illustrate the effectiveness and the advantages of the results obtained.     

Robust periodic stability implies uniform exponential stability of Markovian jump linear systems and random linear ordinary differential equations

In this paper, we mainly show the following two statements.

(1)

A discrete-time topological Markovian jump linear system is uniformly exponentially stable if and only if it is robustly periodically stable, by using a Gel?fand–Berger–Wang formula proved here.     

Razumikhin-type theorems on pth moment boundedness of neutral stochastic functional differential equations with Makovian switching

This paper investigates pth moment boundedness of neutral stochastic functional differential equations with Markovian switching (NSFDEsMS) based on Razumikhin technique and comparison principle. And pth moment stability is examined as a special case. Since the stochastic disturbances and neutral delays are incorporated, the considered system becomes more complex. Besides, the coefficients of the estimated upper bound for the diffusion operation associated with the underlying NSFDEsMS also may be chosen to be sign-changing functions instead of constant functions or negative definite functions, as a result, our results can work in general non-autonomous neutral stochastic systems. Finally, two examples are provided to show the effects of the proposed methods.     

Robust exponential stability of uncertain impulsive delay difference equations with continuous time

In this paper, the robust exponential stability of uncertain impulsive delay difference equations is investigated. First, some robust exponential stability criteria for uncertain impulsive delay difference equations with continuous time in which the state variables on the impulses may relate to the time-varying delays are provided. Then a robust exponential stability result for uncertain linear impulsive delay difference equations with discrete time is given. Some examples, including an example which cannot be studied by the existing results, are also presented to illustrate the effectiveness of the obtained results.     

The Razumikhin approach on general decay stability for neutral stochastic functional differential equations

The topic of the paper is both the pth moment and almost sure stability on a general decay rate for neutral stochastic functional differential equations, by applying the Razumikhin approach. This concept is extended to neutral stochastic differential delay equations. The results obtained in the paper are more general and they may be specialized on the exponential, polynomial or logarithmic stability. Moreover, some neutral stochastic functional differential equations which are not pth moment or almost surely exponentially stable, could be stable with respect to a certain lower decay rate. In that sense, some nontrivial examples are presented to justify and illustrate the usefulness of the theory. More precisely, one can say anything about both the pth moment and almost sure exponential stability, although the solutions are pth moment and almost surely polynomially or logarithmically stable.     

Noise suppresses exponential growth for neutral stochastic differential delay equations

Our aim in this paper is to investigate the polynomial growth for a class of neutral stochastic differential delay systems. We reveal that environmental noise will suppress the exponential growth if it is sufficiently strong. That is, the given system (neutral ordinary differential delay equation) whose solution grows exponentially and becomes a new system (neutral stochastic differential delay equation) whose solution will grow polynomially with probability one. We also provide two examples to illustrate the main result.     

Stability analysis of a class of stochastic differential delay equations with nonlinear impulsive effects

In this paper, we study stability of a class of stochastic differential delay equations with nonlinear impulsive effects. First, we establish the equivalent relation between the stability of this class of stochastic differential delay equations with impulsive effects and that of a corresponding stochastic differential delay equations without impulses. Then, some sufficient conditions ensuring various stabilities of the stochastic differential delay equations with impulsive effects are obtained. Finally, two examples are also discussed to illustrate the efficiency of the obtained results.     

Exponential stability of non-linear neutral stochastic delay differential system with generalized delay-dependent impulsive points

This work aims to analyze the exponential stability of a non-linear impulsive neutral stochastic delay differential system. In this study, impulse perturbation is considered a delay-dependent state variable. The solution of the delay-dependent impulsive neutral stochastic delay differential system is associated with the solution of the system without impulses. First, we developed a relation connecting the solution of the neutral stochastic delay differential system without impulses and the solution of the corresponding system with impulses. Then, the conditions of the exponential stability of the proposed impulsive system are derived by determining the stability analysis of the respective system without impulse. The numerical approach for the neutral stochastic delay system without impulses is generated using the Euler-Maruyama method and adopted for the corresponding impulsive system. Finally, the achieved theoretical results are illustrated for applying the Malthusian single species neutral stochastic delay population model with immigration impulses.     

Further mean-square asymptotic stability of impulsive discrete-time stochastic BAM neural networks with Markovian jumping and multiple time-varying delays

In this paper, the asymptotic stability analysis is investigated for a kind of discrete-time bidirectional associative memory (BAM) neural networks with the existence of perturbations namely, stochastic, Markovian jumping and impulses. Based on the theory of stability, a novel Lyapunov–Krasovskii function is constructed and by utilizing the concept of delay partitioning approach, a new linear-matrix-inequality (LMI) based criterion for the stability of such a system is proposed. Furthermore, the derived sufficient conditions are expressed in the structure of LMI, which can be easily verified by a known software package that guarantees the globally asymptotic stability of the equilibrium point. Eventually, a numerical example with simulation is given to demonstrate the effectiveness and applicability of the proposed method.     

Discrete-time stochastic impulsive BAM neural networks with leakage and mixed time delays: An exponential stability problem

In this paper, the stability analysis of impulsive discrete-time stochastic BAM neural networks with leakage and mixed time delays is investigated via some novel Lyapunov–Krasoviskii functional terms and effective techniques. For the target model, stochastic disturbances are described by Brownian motion. Then the result is further extended to address the problem of robust stability of uncertain discrete-time BAM neural networks. The conditions obtained here are expressed in terms of Linear Matrix Inequalities (LMIs), which can be easily checked by MATLAB LMI control toolbox. Finally, few numerical examples are presented to substantiate the effectiveness of the derived LMI-based stability conditions.     

Optimal control of stochastic functional neutral differential equations with time lag in control

In this work, we consider an optimal control problem of a class of stochastic differential equations driven by additive noise with aftereffect appearing in control. We develop a semigroup theory of the driving deterministic neutral system and identify explicitly the adjoint operator of the corresponding infinitesimal generator. We formulate the time delay equation under consideration into an infinite dimensional stochastic control system without time lag by means of the adjoint theory established. Consequently, we can deal with the associated optimal control problem through the study of a Hamilton–Jacob–Bellman (HJB) equation. Last, we present an example whose optimal control can be explicitly determined to illustrate our theory.     

Dissipativity of discrete-time BAM stochastic neural networks with Markovian switching and impulses

This paper addresses the problem of global exponential dissipativity for a class of uncertain discrete-time BAM stochastic neural networks with time-varying delays, Markovian jumping and impulses. By constructing a proper Lyapunov–Krasovskii functional and combining with linear matrix inequality (LMI) technique, several sufficient conditions are derived for verifying the global exponential dissipativity in the mean square of such stochastic discrete-time BAM neural networks. The derived conditions are established in terms of linear matrix inequalities, which can be easily solved by some available software packages. One important feature presented in our paper is that without employing model transformation and free-weighting matrices our obtained result leads to less conservatism. Additionally, three numerical examples with simulation results are provided to show the effectiveness and usefulness of the obtained result.     

On input-to-state stability of impulsive stochastic systems

This paper is concerned with the input-to-state stability (ISS) of impulsive stochastic systems. First, appropriate concepts of stochastic input-to-state stability (SISS) and pth moment input-to-state stability (p-ISS) for the mentioned systems are introduced. Then, we prove that impulsive stochastic systems possessing SISS-Lyapunov functions are uniformly SISS and p-ISS over a certain class of impulse sequences. As a byproduct, a criterion on the uniform global asymptotic stability in probability for the system in isolation (without inputs) is also derived. Finally, we provide a numerical example to illustrate our results.     

Passivity-based resilient adaptive control for fuzzy stochastic delay systems with Markovian switching

The resilient adaptive controller design problem of a class of Itô-type Takagi–Sugeno (T–S) fuzzy stochastic systems with time-varying delay and Markovian switching is investigated. By utilizing improved matrix decoupling technique, passivity theory and stochastic Lyapunov–Krasovskii functional, LMIs-based sufficient conditions for the existence of resilient adaptive controller are provided such that the corresponding closed-loop system is almost surely asymptotically stable and robustly passive in the sense of expectation. The derived conditions can be easily solved with the help of LMI toolbox in Matlab. A simulation example is presented to illustrate the effectiveness of the proposed resilient adaptive control schemes.