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.     

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.     

Stabilization of time delay systems with saturations via PDE predictor boundary control design

This paper addresses the stabilization issue of linear time delay system with input saturation and distinct input delays via predictor feedback boundary control algorithm by employing transport partial differential equations (PDEs). First, the addressed ordinary differential equation (ODE) system with input delay is equivalently represented as a cascade of an ODE and transport PDEs. Second, by employing the backstepping Volterra integral transformation technique, the equivalent cascade system is transformed into a stable target system, whose kernels are solved by the constraints satisfying transport PDEs. Third, based on the boundary conditions of the obtained invertible transformation, the proposed feedback control law can be formulated. Fourth, by applying semigroup operator theory, the well-posedness of the resulting system is proved and consequently, novel exponential stability conditions of the addressed system are established. Then, the domain of attraction region under the given actuator saturation constraints is estimated with the help of the solution of obtained stability conditions. Finally, a demonstrative simulation example is offered to verify the feasibility and usefulness of the results.     

Stochastic suppression and stabilization of nonlinear differential systems with general decay rate

In this paper, we investigate stochastic suppression and stabilization for a class of non-autonomous differential systems. Given a deterministic non-autonomous differential system, we introduce two independent Brownian motions and perturb this system into a new stochastic differential system. By using Lyapunov analysis method and some stochastic techniques, we show that a polynomial Brownian noise may guarantee the existence of global solution of the perturbed system while another linear Brownian noise may stabilize this system with general decay rate. An application of stochastic stabilization of differential system in the modeling of population growth is indicated.     

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.     

Robust delay-dependent H∞ filtering for uncertain Takagi–Sugeno fuzzy neutral stochastic time-delay systems

This paper addresses the robust H_∞ filter design problem for a class of uncertain fuzzy neutral stochastic system with time-delay through Takagi–Sugeno (T–S) fuzzy model. By constructing an augmented Lyapunov–Krasovskii functional, some novel delay-dependent stability criteria for uncertain fuzzy neutral stochastic system with time varying delay are obtained in terms of linear matrix inequalities. By using the integral inequality in the neutral stochastic setting combined with delay decomposition approach, the H_∞ fuzzy filter is designed to guarantee the corresponding filtering error systems robustly asymptotically stable with a specified H_∞ performance index. At last, two numerical examples are presented to show the less conservatism than the previous results.     

Stabilization of differently structured hybrid neutral stochastic systems by delay feedback control under highly nonlinear condition

This paper mainly studies the stabilization of differently structured highly nonlinear hybrid neutral stochastic systems by delay feedback control. Based on the existing works, our new neutral type stochastic system has completely different highly nonlinear structures in switching subspaces, which is more general and applicable. When such a system is given unstable, we focus on studying the asymptotic and exponential stability criteria by designing a feedback control with a time delay for the underlying system. A simulating example is shown to illustrate the feasibility of these results.     

Stability independent and dependent of delay for delay differential systems

We provide new proofs to modified equivalent conditions for stability independent of delay of retarded and neutral delay differential systems. We also present a new test procedure for stability independent of delay. If the system is not stable independent of delay, the test is further applicable to obtain the intervals of delay for which the system is asymptotically stable. The usefulness and simplicity of the new test procedure is illustrated by numerical examples.     

Properties of value function and existence of viscosity solution of HJB equation for stochastic boundary control problems

In the present paper, we study stochastic boundary control problems where the system dynamics is a controlled stochastic parabolic equation with Neumann boundary control and boundary noise. Under some assumptions, the continuity and differentiability of the value function are proved. We also define a new type of Hamilton–Jacobi–Bellman (HJB) equation and prove that the value function is a viscosity solution of this HJB equation.     

Stability of delayed Hopfield neural networks under a sublinear expectation framework

In this paper, we consider the stability of a class of stochastic delay Hopfield neural networks driven by G-Brownian motion. Under a sublinear expectation framework, we give the definition of exponential stability in mean square and construct some conditions such that the stochastic system is exponentially stable in mean square. Moreover, we also consider the stability of the Euler numerical solution of such equation. Finally, we give an example and its numerical simulation to illustrate our results.     

Dynamical behavior in a hybrid stochastic triple delayed prey predator bioeconomic system with Lévy jumps

In this paper, a hybrid triple delayed prey predator bioeconomic system with prey refuge and Lévy jumps is established, where both maturation delay for prey and predator population and gestation delay for predator population are considered. For deterministic system, positivity and uniform permanence of solution are discussed. Local stability of deterministic system around interior equilibrium is investigated due to variations of triple time delays. For stochastic system without time delay, sufficient conditions for stochastically ultimate boundedness and stochastic permanence are discussed. Existence of stochastic Hopf bifurcation and stochastic stability are investigated. For stochastic system with triple time delays, existence and uniqueness of global positive solution are studied. Finally, combined dynamic effects of triple time delays and Lévy jumps on the hybrid stochastic system are discussed by constructing appropriate Lyapunov functions. Numerical simulations are supported to illustrate theoretical analysis.     

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.     

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.     

Solution of the Stochastic control problem in unbounded domains

Bellman's dynamic programming equation for the optimal index and control law for stochastic control problems is a parabolic or elliptic partial differential equation frequently defined in an unbounded domain. Existing methods of solution require bounded domain approximations, the application of singular perturbation techniques or Monte Carlo simulation procedures.In this paper, using the fact that Poisson impulse noise tends to a Gaussian process under certain limiting conditions, a method which achieves an arbitrarily good approximate solution to the stochastic control problem is given. The method uses the two iterative techniques of successive approximation and quasi-linearization and is inherently more efficient than existing methods of solution.     

Fault diagnosis and fault tolerant control for the non-Gaussian time-delayed stochastic distribution control system

The purpose of fault diagnosis of stochastic distribution control (SDC) systems is to use the measured input and the system output probability density functions (PDFs) to obtain the fault information of the SDC system. When the target PDF is known, the purpose of fault tolerant control of stochastic distribution control system is to make the output PDF still track the given distribution using the fault tolerant controller. However, in practice, time delay may exist in the data (or image) processing, the modeling and transmission phases. When time delay is not considered, the effectiveness of the fault detection, diagnosis and fault tolerant control of stochastic distribution systems will be reduced. In this paper, the rational square-root B-spline is used to approach the output probability density function. In order to diagnose the fault in the dynamic part of such systems, it is then followed by the novel design of a nonlinear neural network observer-based fault diagnosis algorithm. The time delay term will be deleted in the stability proof of the observation error dynamic system. Based on the fault diagnosis information, a new fault tolerant controller based on PI tracking control is designed to make the post-fault probability density function still track the given distribution, which is dependent of the time delay term. Finally, simulations for the particle distribution control problem are given to show the effectiveness of the proposed approach.     

On robust stabilization of uncertain stochastic time-delay systems—an LMI-based approach

In this paper, we first deal with the robust stability of uncertain linear stochastic differential delay systems. The parameter uncertainties are time-varying and unknown but are norm-bounded via two types of uncertainties, and the delays are time invariant. We then extend the proposed theory to discuss the robust stabilization of uncertain stochastic differential delay systems. These results are given in terms of linear matrix inequalities. Two examples are presented to illustrate the effectiveness.     

中立型高阶偏微分方程解的振动性与渐近性

本文研究了一类中立型高阶偏微分方程在第一和第三边值条件下解的振动性质,得到了方程所有解u(x,t)振动或者limt→ ∞乙赘u(x,t)dx=0的一些充分性判别准则。     

The dynamics of an eco-epidemic model with distributed time delay and impulsive control strategy

In this paper, we investigate an eco-epidemic model with distributed time delay and impulsive control strategy. Firstly, by using Floquet theory of impulsive differential equation, we get the condition for the local stability of the prey eradication periodic solutions. Secondly, by means of impulsive equation compare theory, we get the condition for the global asymptotical stability of the prey eradication periodic solutions. Finally we study the permanence of the system. Numerical simulations (bifurcation diagram, the largest Lyapunov exponents and power spectra) are carried out to illustrate the above theoretical analysis and the rich dynamics phenomenon, which are caused by impulsive effects and time delay, for example bifurcation, double period solution, etc.     

Linear quadratic open-loop Stackelberg game for stochastic systems with Poisson jumps

This paper is concerned with the finite horizon linear quadratic (LQ) Stackelberg game for stochastic systems with Poisson jumps under the open-loop information structure. First, the follower solves a LQ stochastic optimal control problem with Poisson jumps. With the aid of an introduced generalized differential Riccati equation with Poisson jumps (GDREP), the sufficient conditions for the optimization of the follower are put forward. Then, the leader faces an optimal control problem for a forward-backward stochastic differential equation with Poisson jumps (FBSDEP). By introducing new state and costate variables, a sufficient condition for the existence and uniqueness of the open-loop Stackelberg strategies is presented in terms of the solvability of two differential Riccati equations and a convexity condition. In addition, the state feedback representation of the open-loop Stackelberg strategies is obtained via the related differential Riccati equation. Finally, two examples shed light on the effectiveness of the obtained results.