Stochastic Differential Delay Systems Research Articles

Stochastic Nicholson-type delay differential system

Focusing on Nicholson-type delay differential system in random environments, we introduce the stochastic system to model the dynamics of Nicholson's blowflies population sizes with mortality rates perturbed by white noises. We study the existence and uniqueness of the global positive solution with nonnegative initial conditions. Then the ultimate boundedness in the mean of a solution is derived under the same condition. Moreover, we estimate the sample Lyapunov exponent of the solution, which is less than a positive vector. Finally, an example with its numerical simulations is carried out to support theoretical results.

Almost sure and moment asymptotic boundedness of stochastic delay differential systems

The purpose of this article is to investigate the almost sure and moment asymptotic boundedness for a class of stochastic delay differential systems. Using stochastic analysis techniques, a generalized non-autonomous L-operator delay differential inequality including cross item is presented. Based on the obtained inequality, some sufficient verifiable criteria for the almost sure and moment asymptotic boundedness of the considered systems are derived analytically. Two numerical examples are given to illustrate the effectiveness of our theoretical results.

Dynamical Analysis and Optimal Harvesting Strategy for a Stochastic Delayed Predator-Prey Competitive System with Lévy Jumps

This paper develops a theoretical framework to investigate optimal harvesting control for stochastic delay differential systems. We first propose a novel stochastic two-predator and one-prey competitive system subject to time delays and Lévy jumps. Then we obtain sufficient conditions for persistence in mean and extinction of three species by using the stochastic qualitative analysis method. Finally, the optimal harvesting effort and the maximum of expectation of sustainable yield (ESY) are derived from Hessian matrix method and optimal harvesting theory of delay differential equations. Moreover, some numerical simulations are given to illustrate the theoretical results.

Boundedness theorems for non-autonomous stochastic delay differential systems driven by [formula omitted]-Brownian motion

This article is devoted to the boundedness problem for a class of non-autonomous stochastic delay differential systems driven by G-Brownian motion. Using theG-Lyapunov function method, some sufficient conditions are derived for the pth moment asymptotical boundedness and stability.

RETRACTED ARTICLE: Existence of weak solutions of stochastic delay differential systems with Schr\xf6dinger\u2013Brownian motions

By using new Schrödinger type inequalities appearing in Jiang and Usó (J. Inequal. Appl. 2016:233, 2016), we study the existence of weak solutions of stochastic delay differential systems with Schrödinger–Brownian motions.

New Criteria on Exponential Stability for Stochastic Delay Differential Systems Based on Vector Lyapunov Function

This paper established the vector Razumikhin-type theorem on exponential stability for stochastic functional differential systems by the stochastic analysis techniques and the property of ${M}$ -Matrix. Several novel stability criteria with vector $ {\mathscr {L}}$ -operator differential inequalities for stochastic delay differential systems, especially some of which includes the cross-item, were obtained by means of the vector Razumikhin theorem. By applying these new results, the robustness of global exponential stability of delay recurrent neural networks to random disturbed has been analyzed.

New Criteria on $p$th Moment Input-to-State Stability of Impulsive Stochastic Delayed Differential Systems

By employing the Razumikhin technique, the uniformly exponentially stable function and the average dwell-time (ADT) approach, some Razumikhin-type theorems on input-to-state stability (ISS) for time-varying impulsive stochastic delay differential systems (ISDDS) are obtained. It is shown that if the time-varying continuous stochastic delayed dynamics is ISS and the impulsive effects are destabilizing, then the ISDDS is input-to-state stable (ISS) with respect to a lower bound of the average dwell-time (ADT). The existing results on ISS of impulsive systems in the literature require the coefficients of the estimated upper bound for the diffusion operator of a Lyapunov function to be constant numbers. While, the results in this note allow us the coefficients of the estimated upper bound of the diffusion operator to be sign-changing time-varying function. An example is given to illustrate the effectiveness of our results. From the example, one can see the existing results fail to be used to determine the ISS for some type of ISDDS.

Exponential stability of hybrid stochastic functional differential systems with delayed impulsive effects: average impulsive interval approach

In this paper, we aim to investigate the exponential stability of general hybrid stochastic functional differential systems with delayed impulses. By using the average impulsive interval and the Lyapunov function method, we derive some sufficient conditions for exponential stability, which are less conservative than those existing results based on the supremum or infimum of impulsive interval and more convenient to be applied than those Razumikhin‐type conditions in the literature. Meanwhile, we show that unstable hybrid stochastic delay differential systems, both linear and nonlinear, can be stabilized by suitably impulsive sequence. Finally, two examples are discussed to illustrate the effectiveness and advantages of the obtained results. Copyright © 2017 John Wiley & Sons, Ltd.

Feedback optimization problem for master–slave teleoperation tracking in the presence of random noise in dynamics and feedback

In this work, we study the problem of a teleoperation master–slave system with error feedback based on sampled versions of the past and present master and slave positions. The sampling interval is the teleoperation delay, and small random fluctuations in the feedback coefficients are considered apart from small random noise in the dynamics. The overall master–slave dynamical system is abstracted into a stochastic delay differential system with random fluctuations in the parameters. This system is transformed into an ordinary stochastic differential system with an infinite-dimensional state vector, and using perturbation theory, integral expressions for the state correlations are obtained in terms of the noise correlations and feedback coefficient fluctuation correlations. We then partially explain how these computations may be used in minimizing the expected value of a cost functional of the state such as the master–slave tracking error energy. Some details about how the underlying delay differential equations for the master–slave dynamics are simulated using the fourth-order Runge–Kutta algorithm are provided. Finally, the simulation experiments are carried out on hardware, i.e., using the actual PHANToM Omni robot and these demonstrate excellent master–slave tracking when sampled teleoperation feedback is given.

Boundedness of Stochastic Delay Differential Systems with Impulsive Control and Impulsive Disturbance

This paper considers thep-moment boundedness of nonlinear impulsive stochastic delay differential systems (ISDDSs). Using the Lyapunov-Razumikhin method and stochastic analysis techniques, we obtain sufficient conditions which guarantee thep-moment boundedness of ISDDSs. Two cases are considered, one is that the stochastic delay differential system (SDDS) may not be bounded, and how an impulsive strategy should be taken to make the SDDS be bounded. The other is that the SDDS is bounded, and an impulsive disturbance appears in this SDDS, then what restrictions on the impulsive disturbance should be adopted to maintain the boundedness of the SDDS. Our results provide sufficient criteria for these two cases. At last, two examples are given to illustrate the correctness of our results.

RobustH∞Control for Linear Stochastic Partial Differential Systems with Time Delay

This paper investigates the problems of robust stochastic mean square exponential stabilization and robustH∞for stochastic partial differential time delay systems. Sufficient conditions for the existence of state feedback controllers are proposed, which ensure mean square exponential stability of the resulting closed-loop system and reduce the effect of the disturbance input on the controlled output to a prescribed level ofH∞performance. A linear matrix inequality approach is employed to design the desired state feedback controllers. An illustrative example is provided to show the usefulness of the proposed technique.

Stability of Stochastic Differential Delay Systems with Delayed Impulses

We investigate the stability of stochastic delay differential systems with delayed impulses by Razumikhin methods. Some criteria on thepth moment and almost sure exponential stability are obtained. It is shown that an unstable stochastic delay system can be successfully stabilized by delayed impulses. Moreover, it is also shown that if a continuous dynamic system is stable, then, under some conditions, the delayed impulses do not destroy the stability of the systems. The effectiveness of the proposed results is illustrated by two examples.

P-Moment Stability of Stochastic Differential Delay Systems with Impulsive Jump and Markovian Switching

This paper investigatesp-moment stability of the stochastic differential delay systems with impulsive jump and Markovian switching. Some stability criteria are obtained based on Lyapunov functional method and stochastic theory. It is shown that, even if all the subsystems governing the continuous dynamics without impulse are not stable, as impulsive and switching signal satisfies a dwell-time upper bound condition, impulses can stabilize the systems in thep-moment stability sense. The opposite situation is also developed for which all the subsystems governing the continuous dynamics arep-moment stable. The results can be easily applied to stochastic systems with arbitrarily large delays. The efficiency of the proposed results is illustrated by two numerical examples.

Optimal estimation of parameters and states in stochastic time-varying systems with time delay

In this study estimation of parameters and states in stochastic linear and nonlinear delay differential systems with time-varying coefficients and constant delay is explored. The approach consists of first employing a continuous time approximation to approximate the stochastic delay differential equation with a set of stochastic ordinary differential equations. Then the problem of parameter estimation in the resulting stochastic differential system is represented as an optimal filtering problem using a state augmentation technique. By adapting the extended Kalman–Bucy filter to the resulting system, the unknown parameters of the time-delayed system are estimated from noise-corrupted, possibly incomplete measurements of the states.

Pth Moment stability of impulsive stochastic delay differential systems with Markovian switching

This paper is concerned with the pth moment stability of impulsive stochastic delay differential systems with Markovian switching. By using the Razumikhin-type method, some stability criteria are obtained, which can loosen the constraints of the existing results and thus reduce the conservativeness. Two examples are presented to demonstrate the usefulness of the proposed results.

Delay-dependent exponential stability of stochastic delay differential system whose coefficients obey the polynomial growth condition

This article discusses the exponential stability of nonlinear stochastic delay differential systems (SDDSs) whose coefficients obey the polynomial growth condition. Delay-dependent criteria on almost sure exponential stability and pth moment exponential stability of such SDDSs have been established. By applying some novel techniques, our criteria work for many SDDSs including some cases in which the ℒV operator has a complicated form, which seemingly prevents the existing results from being directly used. The range of order of moment exponential stability and the decay rate can be estimated through the coefficients of the system.

The stability of distributed neutral delay differential systems with Markovian switching

This paper is concerned with the stability analysis of neutral-type stochastic distributed delay differential systems described by Markovian switching. This system has some special kind of neutral behaviour with uncertain distributed time delays occurring in the state variables. Based on the Lyapunov function, novel methodologies for analyzing stability criteria, and the design of an uncertain distributed delay model are presented. The proposed method is an alternative way to study the robustness and stability of uncertain distributed delays with neutral systems. In order to demonstrate the applicability of the results, the investigation considers two specific examples.

Exponential Stability of Impulsive Stochastic Delay Differential Systems

This paper investigates the stability of stochastic delay differential systems with two kinds of impulses, that is, destabilizing impulses and stabilizing impulses. Both the pth moment and almost sure exponential stability criteria are established by using the average impulsive interval. When the impulses are regarded as disturbances, a lower bound of average impulsive interval is obtained; it means that the impulses should not happen too frequently. On the other hand, when the impulses are used to stabilize the system, an upper bound of average impulsive interval is derived; namely, enough impulses are needed to stabilize the system. The effectiveness of the proposed results is illustrated by two examples.

Exponential Stability of Impulsive Stochastic Functional Differential Systems

This paper is concerned with stabilization of impulsive stochastic delay differential systems. Based on the Razumikhin techniques and Lyapunov functions, several criteria on pth moment and almost sure exponential stability are established. Our results show that stochastic functional differential systems may be exponentially stabilized by impulses.

Exponential p-stability of singularly perturbed impulsive stochastic delay differential systems

In this paper, we study singularly perturbed impulsive stochastic delay differential systems (SPISDDSs). By establishing an L-operator delay differential inequality and using the stochastic analysis technique, we obtain some sufficient conditions ensuring the exponential p-stability of any solution of SPISDDSs for sufficiently small ɛ > 0. The results extend and improve the earlier publications. An example is also discussed to illustrate the efficiency of the obtained results.