The Applications Of LMI Approach To Stochastic Delay Differential Equations

If the state of a system is caused by numerus factors which influence on the system in a stochastic way, then environment noise should be taken into account in the description of the system. In addition, when the development trend of this system is related with not only present state but also history or future state, the rule of this kind of systems is often described by stochastic delay differential equation. With the deep development of the science study, the description on practical systems is required to be better and better. To satisfy the requirement, some more special stochastic differential equations, such as stochastic delay differential equations with parameter uncertainty or Markovian Switching, singular stochastic delay differential equations etc. provide a new mathematical tool, which have widely applied in the domain of control theory, artificial intelligence, network theory, biology theory, finance etc. Among the stochastic differential equations theory, two basic problems are the estimations and stability of the solutions. Therefore, we shall focus on the study of the two problems on nonsingular and singular stochastic delay differential equations with parameter uncertainty or Markovian switching in this thesis.An LMI approach is an important technique to study the asymptotic behavior and stability of the solutions of a system, because the use of algebraic measures in the analysis has the benefits of being simple computationally and more applicable. In actual engineering application, the LMI approach is widely used, especially in the analysis and synthesis for uncertain systems. Therefore, we focus on how the LMI approach is applied to deals with the problems for the estimations and stability of the solutions for stochastic delay differential equations.By means of Lyapunov-Krasovskii stability theory on functional differential equations,Ito equation and generalized Ito equation, as well as Gronwall inequality and Schur complement inequality, the LMI-based estimate conditions are obtained for a class of stochastic delay differential equation with parameter uncertainty in this thesis, and hence estimates of the solutions are determined by the LMIs above. Also, we discuss the estimates of the solutions for neutral stochastic delay differential equations with parameter uncertainty basing on LMI approach. Meanwhile, the estimation of the solution of stochastic delay differential equations with Markov switching and parameter uncertainty is firstly investigated by constructing new Lyapunov-Krasovskii functional, and the estimates can be achieved by the solutions of a group of LMIs. Finally, some numerical examples is given to illustrate the efficiency of LMI approach on the problem of the estimates for this class of stochastic delay differential equations. And our conclusions include and extend the others' results in existing references for uncertain differential equations without stochastic term.The stability of a class of nonlinear stochastic delay differential equation with Markovian switching is studied. Basing on Lyapunov-Krasovskii functional stability theory and modern probability theory, the thesis establishes delay-independent criterions on stability of the hybrid systems, where the sufficient conditions for stability is presented in terms with LMIs. Meanwhile, by constructing an equivalent augmented system, that is, a singular form representation of the system, new delay-dependent stability criterion is developed. And each of the two criterions has its strong points. Finally, some examples are provided to illustrate the applicability of the presented conclusions. Our results generalize some of existing results in the references.Further on, the stability problem of singular stochastic delay differential equations with Markovian switching is investigated. And the delay-independent and delay-dependent stability criterions are established via the similar analysis for nonsingular stochastic delay differential equations. To the best knowledge of author, the known results of delay-dependent stability criterions are for deterministic nonsingular differential equations or linear stochastic singular differential equations , there are no such results for nonlinear stochastic singular differential equations.In addition, LMI approach has often been adopted to study the stability of deterministic neural network. And in practical neural systems, neural signal transmission is a process filled with noise and always influenced by stochastic factors, also time delay greatly influences on performance of neural network. To describe neural network properly and then design, analysis and apply it, it is required to introduce noise and time delay into neural network. In this thesis, we obtain the sufficient conditions for the stability of multi-delayed stochastic neural network by using LMI approach. The conditions take the form os LMIs, so they are verifiable and computable efficiently. In addition, our results are established without restricting differentiability and monotonicity of the activation functions, hence our model is a more general model. The main results in this thesis are the generalization or extension of some recent results in the literature.