Research On Stability And Numerical Methods Of Nonlinear Or Delayed Stochastic Systems

Stochastic perturbations exist inevitably in the real systems and the external environ-ments, which infuence the dynamical behavior of systems. As stochastic systems depictedby stochastic diferential equations (SDEs) can simulate practical problems truthfully andrefect the dynamical characteristics of natural, social and engineering systems more ac-curately, they have been widely applied to model the corresponding systems in manyfelds such as engineering applications, mathematical fnance, neural networks, ecology,medicine and control science. The stability and control theory of stochastic systemswith various complicated factors such as nonlinearities, time-delays, varying coefcients,Markov jumps and distributed parameters are the current research hotspots. Because ofthe difculty in obtaining the explicit solutions of nonlinear or delayed SDEs, it is animportant subject both in theoretical signifcance and practical application to constructappropriate numerical algorithm for the simulation to the solutions. This doctoral dis-sertation aims at the stability analysis and numerical study for the nonlinear or delayedstochastic systems.The main contributions of the dissertation are summarized as follows:1. The background, research signifcance and current situation of the selected topicare reviewed. Some preliminaries are also presented.2. The problem of exponential stability in mean square is investigated for a class ofuncertain stochastic systems with multiple delays and nonlinear perturbations. By em-ploying some free weighting matrices and constructing Lyapunov-Krasovskii functional, asufcient condition of delay-dependent robust stability is derived in terms of linear matrixinequalities(LMIs). Finally, an example is provided to demonstrate the efectiveness ofthe proposed method.3. The mean-square exponential input-to-state stability (exp-ISS) of Euler-Maruyama(EM) method to stochastic delay control systems(SDCS) is developed. The study of ISSusually depends on the existence of an appropriate Lyapunov function or functional.However, there is no an efective method to fnd such Lyapunov function or functional.The numerical method provides a new method for studying the characteristics of controlsystems and solving the practical problems. The conditions of the exact and EM nu-merical solutions to an SDCS having the property of mean-square exp-ISS are obtainedwithout involving control Lyapunov functions or functional. Supposing the coefcientsare global Lipschitz coefcients and the inputs are mean-square continuous measurable, it is proved that the mean-square exp-ISS of an SDCS holds if and only if the mean-squareexp-ISS of the EM method is preserved for sufciently small step size. The proposedresults are illustrated by numerical experiments.4. The general decay stability of stochastic diferential equations with Markovianswitching is investigated by employing more general decay functions. Firstly, some-stability criteria in p-th moment and almost surely sense for the analytical solutionsare established, by utilizing Ito formula, Borel-Cantellion and martingale exponentialinequalities. Then the Euler-Maruyama method is shown to capture-stability behaviorfor all sufciently small timesteps under appropriate conditions.5. The convergence and stability of Milstein method for nonlinear stochastic de-lay integro-diferential equations are considered. The Milstein schemes are developedfor nonlinear SDIDEs with both discrete lag and distributed lag by applying the multi-dimensional Ito formula combining Taylor expansion in the frst time. Based on the rela-tion of consistency and convergence, multiple Ito integral properties and some stochasticanalysis techniques, the Milstein method converging with strong order p=1.0in themean-square sense is proved. The sufcient condition ensuring the the exponential sta-bility in mean-square of the proposed method is obtained and numerical experiments arepresented.6. The exponential stability in mean square sense of a class of stochastic delayrecurrent neural networks is investigated in depth. Firstly, by using Ito formula andinequality techniques, the sufcient conditions guaranteing the exponential stability inmean square of an equilibrium are given. Then, the stability property of the system isdiscussed further by using EM method and split-step backward Euler (SSBE) methodrespectively. At last, an example is given to demonstrate our results and compare thetwo methods.Finally, the main results of the dissertation are concluded and some issues for futureresearch are proposed.