Research On Stability Of Stochastic Delay Differential Equations

As an important mathematic model, stochastic delay differential equations is deter-mine delay differential equations with random elements or stochastic differential equa-tions with time delays. That is, stochastic delay differential equations can model realproblems specifically. Therefore, they have been widely applied in many fields of sci-ence, such as automatic control, neural networks, biology, economics, chemical reactionengineering etc. Due to the instability comes from stochastic disturbance and delay ef-fects, It is very necessary to study the stability of stochastic delay differential systems.The research backgrounds and research status on stability of stochastic delay differ-ential equations are introduced. It also provides the readers with some basic definitionsand important lemmas that are frequently used in this dissertation.Firstly, the exponential p-stability for linear neutral stochastic differential equationswith variable delays, the exponential p-stability and asymptotically p-stability for linearneutral stochastic differential equations with mixed delays are studied respectively. Byconstructing suitable contraction mapping and employing fixed point theory, applying Cpinequality, Burkholder-Davids-Gundy inequality, Holder inequality, some novel stabilitycriteria are derived.Secondly, the bounded-input bounded-output(BIBO) stability in mean square forstochastic differential equations with nonlinear perturbation and delays (discrete time de-lays and mixed time delays) is investigated. Using Razumikhin technique and comparisonprinciple to obtain the novel BIBO stabilization criteria in mean square for stochastic de-lay differential systems with nonlinear perturbation, based on this, the design of statefeedback is given by matrix transform. Combine Lyapunov function theory with Ric-cati equations to analyze controller, some delay-dependent stability criteria for stochasticdifferential equations with nonlinear perturbation and mixed delays are obtained. Makegood use of inequality and Lyapunov function theory, some novel delay-dependent BIBOstability criteria in mean square are derived and formulated in the form of linear matrixinequalities (LMIs) by constructing a new class of Lyapunov-Krasovskii functionals andthe descriptor model of the system and the method of decomposition.Lastly, the stability of discrete-time stochastic neural networks(DSNNs) with de- lays(mixed time delays and discrete random time delays and distributed time delays)is studied. For DSNNs with mixed time delays, based on delay partitioning idea andfree-weighting matrix approach, applying Schur complement, a less conservative delay-dependent asymptotically stable criterion in the mean square can be developed. InDSNNs, the time delay is assumed to be Bernoulli stochastic vary and always ap-pears in random way. Then the information of the probability distribution of the time-varying delay is considered and transformed into parameter matrices of the transferredDSNN model, by constructing a new augmented Lyapunov-Krasovskii functional andintroducing some new analysis techniques, some delay-probability-distribution- depen-dent robust exponential stability criteria in mean square for DSNNS with randomlytime-varying delays and parameter uncertainties are derived, some delay-probability-distribution-dependent asymptotically stability criteria in mean square for DSNNS withdiscrete randomly time delays and distributed time delays are derived.We summarize the main results obtained in this dissertation, and point out the futureworks that have been the author's concerns.