Research On Stability Of Nonlinear Hybrid Stochastic Differential Delay Equations

In practical systems,many stochastic dynamical systems are not only disturbed by some random factors,such as Brownian motion,Lévy noise and colored noise,etc.,but also may encounter some abrupt phenomena,which may cause some random sudden changes in parameters or structures of the system.For purpose of describing this kind of systems more accurately,hybrid stochastic differential equations are presented,where stochastic differential equations are applied to characterize the continuous variation of the system state,as well as the Markov chain with finite state is applied to characterize the effect of random mutations on the variation of the system state.Additionally,the evolvement of the practical system is not only determined via the present state of the system,but also related to the past state.In consequence,it is necessary to take into account the influence of the time delay on hybrid stochastic differential equations.In fact,the stability issues of hybrid stochastic differential delay equations have always been one of the momentous research subjects in academia.This paper makes primarily some discussion regarding the stability of nonlinear hybrid stochastic differential delay equations disturbed by Brownian motion,Lévy noise and colored noise.Provide the demonstration of the existence and uniqueness of the global solution.Propose several sufficient criteria regarding the stability of the global solution.Perfect theoretical results regarding the stability of nonlinear hybrid stochastic differential delay equations.The primary research contents are generalised as below:The asymptotic stability problem is solved for a class of nonlinear hybrid stochastic differential delay equations driven by Brownian motion.Suppose that the system has different parameters under different modes and the considered state time delay is variable delay with the property of random occurrence,which is depicted via a sequence of stochastic variables subject to the Bernoulli distribution.For purpose of reducing the conservativeness of the delay-independent stability criterion arisen from insufficient consideration of the time-delay information,a Lyapunov-Krasovskii functional with a double integral term is designed via applying the Lyapunov stability theorem of stochastic systems and the upper bound of the variable delay.A relatively weak nonlinear hypothesis condition is proposed,which is combined with the local Lipschitz condition and the linear growth condition,to demonstrate the existence and uniqueness of the global solution by using the M-matrix property.Several delay-dependent sufficient criteria are established for guaranteeing the asymptotic stability in p-th moment,the almost surely asymptotic stability,the continuity in p-th moment as well as the continuity in probability of the global solution.A couple of numerical examples are utilized to verify the feasibility of the presented theory findings.The applicability of the proposed delay-dependent stability criteria on partly unknown transition rates of the Markov chain is analyzed.The exponential stability and asymptotic stability problems are discussed for a class of nonlinear hybrid stochastic differential delay equations driven by Brownian motion.Suppose that the system has different structures under different modes and the considered time delay is variable delay.In the light of the relevant information of the bound of the time delay and the order of the nonlinear function,a Lyapunov-Krasovskii functional with a double integral term and a higher-order nonlinear term is designed to solve difficulties arisen from different nonlinear structures under different modes and the time delay.A sufficient condition is provided to show the existence and uniqueness of the global solution on the basis that nonlinear functions satisfy the local Lipschitz condition,the polynomial growth condition as well as the nonlinear growth condition.Several delay-dependent sufficient criteria are established for guaranteeing the asymptotic boundedness in p-th moment,the exponential stability in p-th moment,the asymptotic stability in p-th moment as well as the almost surely asymptotic stability of the global solution.The feasibility of the presented theory findings and the superiority of the delay-dependent stability criteria are demonstrated via a couple of numerical examples.The exponential stability problem is investigated for a class of nonlinear hybrid stochastic differential delay equations disturbed by Lévy noise.Suppose that the system has different structures under different modes and the considered time delay is variable delay.A mode-dependent Lyapunov function related to the order of the nonlinear function is designed to solve difficulties caused by different nonlinear structures and the different nonlinear growth under different modes.Without the linear growth condition,a sufficient criterion is provided for illustrating the existence and uniqueness as well as the asymptotic boundedness in p-th moment of the global solution via combining the M-matrix property.A sufficient criterion is obtained for ensuring the exponential stability in p-th moment as well as almost surely exponential stability of the trivial solution via using the non-negative semi-martingale convergence theorem.An approach to solving the admissible the upper bound of the time-delay derivative is given for guaranteeing the asymptotic boundedness and the exponential stability the global solution.A couple of numerical examples are utilized to show the feasibility of the presented theory findings.The characteristics of Lévy noise,Poisson jump and Brownian motion and the variation trend of homologous solutions are analyzed based on state curves of the system.The noise-to-state stability and asymptotic stability problems are discussed for a class of nonlinear hybrid stochastic differential delay equations disturbed by colored noise.Suppose that the considered time delay is multiple time-varying delays and the considered colored noise is mean-square finite or mean-square bounded.A new hypothesis condition based on quasi polynomial is proposed,which is combined with the local Lipschitz condition and the polynomial growth condition,to demonstrate the existence and uniqueness of the global solution.Moreover,sufficient criteria are established for guaranteeing the asymptotic boundedness in q-th moment and the noise-to-state stability in q-th moment of the global solution.Further,based on the case that the colored noise is bounded in the mean square sense,several delay-dependent sufficient criteria are given for ensuring the asymptotic stability in p-th moment as well as the almost surely asymptotic stability of the global solution.A couple of numerical examples with different colored noises are used to show the feasibility of the proposed theory findings and the influence of the upper bound of the colored noise on the stability.The colored noises considered in our numerical examples are the random disturbance driven by Brownian motion and the random disturbance driven by random variables obeying uniform distribution,respectively.