Stochastic Stabilization Of Dynamic Systems Driven By Reflecting Brownian Motion

In the real environment, the system always works under all kinds of internal and external disturb, and there are no systems that are not completely disturbed by random noise.We must pay more attention to the operation of systems that are disturbed by various noise. After bearing this interference, it is very important to make sure that the disturbed systems continue to work and don’t lose control. So the stability of dynamic systems is very meaningful in theory and practice. In this paper, we mainly study almost sure exponential stability of a kind of continuous semi-martingale stochastic differential equations and the stochastic stabilization of an unstable linear system driven by random noise that is a reflecting Brownian motion.In the first chapter, we mainly discuss the research significance, methods and achievements of stability of the stochastic differential equation, and then give a simple introduction to the main content and research work of this article. In the second chapter, we mainly introduce some basic concepts and theorems about the continuous semi-martingale, continuous semi-martingale stochastic differential equations, conditions for the existence and uniqueness of solutions to continuous semi-martingale stochastic differential equations, and the definition and judgment theorem of almost sure exponential stability of Ito stochastic differential equation. In the third chapter, we consider the almost sure exponential stability of a class of semi-martingale stochastic differential equations. According to the Lyapunov direct method and continuous semi-martingale Ito ’s formula, we get sufficient criterion of the almost sure exponential stability of this kind of continuous semi-martingale stochastic differential equation and give the proof.Under certain conditions, Brownian motion can stabilize a given dynamical system. As a transformation of Brownian motion, reflecting Brownian motion keeps a way of Brownian motion in certain region. The fourth chapter studies the stochastic stabilization of an unstable linear system driven by random noise that is a reflecting Brownian motion. Firstly, the development results of stochastic stabilization are introduced, and then we review the theory of stochastic stabilization of nonlinear systems driven by Brownian motion and introduce the reflecting Brownian motion. Finally, we consider an unstable linear system, add the reflecting Brownian motion noise to the system, and give the condition of exponential stability of the disturbed and an example is illustrated.