Stability Of Several Classes Of Stochastic Systems And Their Applications

Since the concept of stochastic integral was introduced by It ?o, stochastic systemstheory as a new branch of mathematics was rapidly developed. As the mathematical mod-el of stochastic systems, the stochastic differential equation plays an important role andhas been widely used in many fields such as physics, chemistry, mechanics, economicsand finance systems, aerospace engineering systems and control theory. With the develop-ment of the stochastic systems theory, the content of stability theory is also expanded andimproved continuously. The stability is an indispensable premise to keep the stochasticsystem operating normally, so the stability of stochastic systems is a challenging problemand has a broad prospect of application. It is difficult to get the explicit solutions for moststochastic differential equations. Therefore, it is very valuable to construct the propernumerical methods to solve stochastic differential equations in practical applications. Inthis dissertation, several classes of stochastic systems are studied, and the stability of thestochastic systems and the numerical methods are discussed.The main contents are as follows:This dissertation first describes the background and research significance of stochas-tic systems, also the situation of the development and the basic theories of stochasticsystems are reviewed. We present the related basic knowledge of the stochastic systemstheory and stability theory, and briefly introduce the main work.The robust stability for a class of continuous time stochastic systems containingnonlinear perturbations is studied. The static output feedback controllers are designed torobustly stabilize the systems. Combined with the stability of the stochastic differentialequations, and with the use of the linear matrix inequalities, the problems are converted tothe convex optimization problems with linear matrix inequalities constraints, which canmake them easy to be solved numerically. Finally, a numerical example is given to verifythe effectiveness of the approach of the linear matrix inequalities.The numerical scheme and its stability for a class of stochastic systems with ran-dom noise at the right side of the system are discussed. The mathematical models ofthe stochastic systems are described by stochastic partial differential equations. The highaccuracy compact finite difference scheme is constructed to solve the equations, also theerror analysis and the stability analysis which through the matrix theory are given. Fur-ther, by using the computer numerical simulation, we can get that the accuracy of thisscheme is higher than the other two schemes.A class of stochastic systems with random parameters is analyzed. The orthogonalpolynomial method is used to transform this kind of stochastic system to its equivalentdeterministic system. Through the computer simulation, the Poincar′e maps and phaseportraits of the random parameters system and its mean system are plotted, which showthat the two systems have similar dynamic behaviors. Then, the stochastic averagingmethod for quasi Hamilton system and stabilization strategy are used to study the delayedfeedback control under colored noise excitation of the mean systems, and the numeri-cal simulation illustrates that the proper control of noise intensity which can effectivelyrealize the stabilization of the system.The stabilization control of a class of marine power systems with random oscillationis investigated. According to the Lyapunov stability theory, the original system will ap-pear chaotic oscillation under the disturbance of random factors. Further, by computingthe top Lyapunov exponent of system, we can get the stabilization of the system by addinga very small intensity noise to the phase of system. The numerical simulation results canfurther verify the obtained conclusions.