Research On Stability, Stabilization And Control Of Nonlinear Or Delayed Stochastic Systems

There always exist random factors in the real systems and their external environments, which influence the dynamical behavior of the systems. In fact, sometimes, random models can more accurately reflect the dynamical characteristics of natural, social and engineering systems. At the time being, the control theory for stochastic systems with various complicated factors such as nonlinearities, time-delays, varying coefficients,Markov jumps, impulses, distributed parameters and fuzziness are the current research hot-spots. In this dissertation, the stability, stabilization and control problems will be investigated for the nonlinear or delayed stochastic systems. Novel methods and approaches for the stability analysis, state feedback stabilization and stabilization by noise will be developed in order that less conservativeness and more features on randomness are implied by the stability criteria obtained. The methods for stability analysis by moment equations for nonlinear stochastic systems and those for stability analysis by Lyapunov functions combined with the system equations for the delayed stochastic systems will be mainly explored, and the Lyapunov stability theorem with more features on randomness,new type Razumikhin theorem and the operator approach for the stability theorem of delayed stochastic systems will be also established. We will bring some new visions and theoretical methods for the classic stability analysis and stabilization problems of nonlinear or delayed stochastic systems, which will further improve and develop the theory for stochastic systems and provide theoretical basis for the engineering and social practices.The main contributions of the dissertation are summarized as follows:1. The background, research significance and current situation of the selected topic are reviewed. Some preliminaries are also presented, including the most used notations,some related lemmas, definitions and theorems. In addition, the basic method for numerical simulation for the selected topic is presented, and an interesting exploration with the research method for the functional differential equations based on Lyapunov functions are introduced briefly. Lemma 1.8 with its corollary, the numerical scheme and the exploration on the Lyapunov function method are all new results obtained in this dissertation.2. Moment stability of nonlinear stochastic systems with time-delays, including the continuous and discrete systems, is investigated. Based on the Kronecker algebra and the H-representation technique, we obtain the second order moment equations of nonlinear stochastic systems with time-delays. By the comparison principle and the established moment equations, we get the comparative systems of the nonlinear stochastic systems with time-delays. Then we establish moment stability theorems for the systems with the stability properties of the comparative systems. Finally, an example is presented to illustrate the efficiency of our results.3. Based on the Lyapunov function method, new type stability criteria for It?o stochastic functional differential equations is developed. Firstly, the concepts “freezing operator” and “quasi-negative definiteness of stochastic derivatives” are proposed. Based on the freezing operator and generalized differential inequalities, we have established a kind of new type stability criteria for It?o stochastic functional differential, which admits weakened assumptions and is in a general form. In addition, the conclusion can be degenerated to deterministic functional differential equations, and the employed Lyapunov function method can be generalized to the multiple Lyapunov functions method.4. The functional differential inequalities is further investigated. Based on the comparison principle established firstly in this part, we generalize the usual ordinary differential inequalities to the corresponding functional differential inequalities, which are considered for the arbitrary delays, including the infinite ones. As the results, we generalize a classical functional differential inequality i.e. the Halanay inequality to the nonlinear case with arbitrary delays and the time-varying linear case. As an application,we investigate the stability of time-varying It?o stochastic systems with distributed delays,and obtain a stability criterion based on the results obtained on functional differential inequalities. Finally, an example is presented to illustrate the efficiency of our results.5. A new type stability theorem for stochastic functional differential equations(SFDEs) is established, which is not a direct copy of the basic stability theorem for deterministic functional differential equations(DFDEs). By the new type stability theorem,one can use the most simple Lyapunov functions and employ the equations repeatedly to deal with the delayed terms encountered conveniently and to carry out stability criteria for the equations. Based on the theorem, a practical stability theorem in accordance with the Lyapunov function method, is also established, and then the asymptotic stability of SFDEs with distributed delays in the diffusive terms is investigated and a stability criteria for SFDSs is obtained, which is described by algebraic matrix equations. Finally, an example is given to illustrate the effectiveness of our method and results.6. Operator-type stability theorems are established. Based on the generalized differential inequalities mentioned above, asymptotic stability of general retarded stochastic systems is investigated. Firstly, the method for rewriting system models via functional difference operators is proposed, and two asymptotic stability theorems based on functional difference operators with the Lyapunov functional method and Lyapunov function method are established respectively, which are in general forms suitable for neutral systems and convenient for applications. At the last part, as an application of the method,stabilization of time varying linear stochastic systems with distributed delay, especially in the diffusive terms, is investigated, the design method for the control law is illustrated and the corresponding criterion is given. Finally, an example is presented to illustrate the efficiency of our results.7. The concept of Razumikhin-type functional differential inequalities is explicitly proposed. Based on Razumikhin-type functional differential inequalities, the comparison principle for Razumikhin-type functional differential inequalities is established, and then the quantitative properties implied by Razumikhin-type functional differential inequalities are investigated via the established comparison principle. As a direct application, based on the established comparison principle, some novel Razumikhin-type stability theorems for the deterministic and stochastic systems are established respectively. Finally, an example is given to illustrate the application and the efficiency of the proposed method.8. Divided state feedback control of stochastic systems is investigated. Firstly, the concepts of state extraction matrix and divided feedback control are proposed. Secondly,a fine stability criterion is established for stochastic systems with dominating linear parts.Thirdly, the divided state feedback control of delayed stochastic systems is investigated,the divided state feedback control law is designed and the corresponding stability criterion for the closed-loop system is established. Finally, fault tolerant control is investigated facing to the cases with partial state information lost or delivered too late caused by the processes for sampling and signal transmission via networks. At the end of this part, an example is given to illustrate the application and efficiency of the proposed method and the advantage of the divided feedback control.9. A kind of new type stability theorems for stochastic systems, including continuous and discrete systems, are established, which are of LaSalle type theorems actually. For continuous and discrete stochastic systems, stochastic stabilization and destabilization by noise are further investigated based on these stability theorems. In this part, the local Lipschitz condition is weaken to the generalized local Lipschitz condition, which admits variable coefficients. The linear growth condition or one-side linear growth condition used in the previous literature is weaken as the generalized one-side linear growth condition,which is local, variable and nonlinear, admits real variability in the time argument and nonlinearity in the state argument. As an application of the new type stability theorems,1. a simple and direct design method is proposed for finding a noise strength ?g(t, x) so that the noise ?g(t, x)d?B(t) stabilizes an unstable system or destabilizes a stable system,deterministic or stochastic. The design method is suitable for the variable and nonlinear systems. 2. Against the background of memristor-based circuits, stochastic stabilization and destabilization for discontinuous systems is investigated. The generalized It?o’s rule,the non-zero property and global existence of the Filippov solutions of stochastic systems with discontinuous drafts are stated briefly. Based on the same design method proposed for the continuous systems, stochastic stabilization method for memristor-based circuits is investigated. The stochastic stabilization method for memristor-based circuits provides global “controllers” without any restrictive condition on the parameters of the systems as well as any switching. At the end, several illustrative examples are presented to show the usage and the efficiency of the proposed method.The feature of this dissertation lies in: by careful observations on the related problems and employed methods, via the combination of various research methods and based on some novel methods we proposed, aiming at the difficulties brought about by the randomness, nonlinearity, delays and time variance in the systems, it launched a new round of investigations, around the classical difficult problems in the our direction, and tried to achieve some breakthroughs with the the models, problems and assumptions which were difficult to be developed, investigated and weakened respectively in the past time. The author thinks that the methods proposed and results obtained in this dissertation are all elementary, but an enlightenment has been obtained by the serial explorations in the dissertation, that is, one may miss precious wealth when we go around difficulties without forethought, thus the author wishes to continue the investigations of the dissertation and hopes to achieve new developments in the future. The directions for the further investigations will be summarized in “The outlook” at the end of the dissertation.