Stability, Controllability Of Infinite Dimensional Stochastic Systems With Applications

Stochastic perturbation is inevitable in any real world system. To describe the real systemsmore exactly and further to design better controllers, it is necessary to take account of stochasticfactors. Infinite dimensional stochastic systems are described by stochastic partial differentialequations, stochastic integral equations or stochastic evolution equations in abstract spaces. Itsmain characters are being in?uenced by stochastic noises and the state variables belonging toinfinite dimensional function space. Nowadays, infinite dimensional stochastic systems havebeen widely applied in many fields, such as modern quantum mechanics, hydromechanics,ocean atmosphere forecasting, image processing, industrial control, economics, biology, etc.Infinite dimensional stochastic systems and their control theory have become a hot spot inmodern control theory and mathematics.Based on the theory of infinite dimensional stochastic analysis, linear operator semigroup,and modern partial differential equations, this dissertation explores some new analysis tech-nique and systematically studies the stability and controllability of infinite dimensional stochas-tic systems by means of Lyapunov method, Banach fixed point theorem and linear operatorinequality. Some important theoretical and practical results are obtained. The main contentsand contribution of this dissertation are summarized as follows:1. An introduction to the latest progress and research method in stability and controlla-bility of infinite dimensional stochastic systems is given. Also, stochastic integral in infinitedimensional spaces and its properties, some important inequalities are presented. And then theIt(o|^) formula for infinite dimensional stochastic partial differential equations is given.2. The existence and uniqueness of the solution of stochastic hyperbolic equation withnon-Lipschitz drift and diffuse coefficients are discussed. By using Green function presentationof mild solution and the formula for the variation of parameters, approximate sequence of mildsolution are constructed in some suitable space and proven converging to the real solution.The obtained existence and uniqueness theorem is typical. It is easy to find that the Lipschitzcondition is only a special case of our result. 3. The complete controllability problem of semilinear impulsive stochastic systems ininfinite dimensional space is concerned by using Banach fixed point theorems. Utilizing theformula for the variation of parameters, B-D-G inequality and introducing feedback control,the systems are proven to be completely controllable in certain small interval. An example isgiven to illustrate the theorem.4.Stability of first order nonlinear stochastic hyperbolic systems in whole space is con-cerned. Employing Fourier transformation and C0 semigroup theory, sufficient conditions en-suring exponentially stable in mean square and almost surely exponentially stable are givenunder strong hyperbolic assumption. An example is provided to illustrate our theory.5. The stability of a class of linear stochastic delay systems in infinite dimensional space isconsidered. The system delay is admitted to be unknown, time-varying, multi delays and multivarying delays, but the operator acting on the delayed states is bounded. Sufficient conditionsensuring exponential stability of abstract infinite dimensional stochastic time delay systems atgiven decaying rate are derived in the form of LOI. We point out the LOI in infinite dimensionalspaces is a generalization of LMI in finite dimensional spaces.6. Based on linear operator inequality, the stability of stochastic heat equations andstochastic wave equations with time delay is considered. For stochastic delay heat equationswith constant coefficients, the in?uence of delay and noise to the mean square exponential sta-bility is discussed by Green formula and Poincare′inequality. Noting the conversation of energy,stability of one order linear stochastic delay wave equation with dissipation term is considered.By constructing Lyapunov functions, sufficient conditions ensuring mean square exponentialstability of strong solution is obtained in the form of LMI.7. The boundary stabilization and robust H_∞problem of semilinear stochastic parabolicequations are considered. By designing boundary static state controller, sufficient conditions ofmean square exponential stability and robust stabilizations are derived in the form of LMI.Finally, the main results of the dissertation are summarized, and the issues of future inves-tigation are proposed.