#### Access Type

Open Access Dissertation

#### Date of Award

January 2014

#### Degree Type

Dissertation

#### Degree Name

Ph.D.

#### Department

Mathematics

#### First Advisor

Gang G. Yin

#### Abstract

This dissertation concerns the properties of nonlinear dynamic systems hybrid with Markov switching. It contains two parts. The first part focus on the mean-field models with state-dependent regime switching, and the second part focus on the system regularization and stabilization using feedback control. Throughout this dissertation, Markov switching processes are used to describe the randomness caused by discrete events, like sudden environment change or other uncertainty.

In Chapter 2, the mean-field models we studied are formulated by nonlinear stochastic differential equations hybrid with state-dependent regime switching. It originates from the phase transition problem in statistical physics. The mean-field term is used to describe the complex interactions between multi-bodies in the system, and acts as an mean reversing effects. We studied the basic properties of such models, including regularity, non-negativity, finite moments, existence of moment generating functions, continuity of sample path, positive recurrence, long-time behavior. We also proved that when switching process changes much more frequently, the two-time-scale limit exists.

In Chapter 3 and Chapter 4, we consider the feedback control for stabilization of nonlinear dynamic systems. Chapter 3 focus on nonlinear deterministic systems with switching. Many nonlinear systems would explode in finite time. We found that Brownian motion noise can be used as feedback control to stabilize such systems. To do so, we can use one nonlinear feedback noise term to suppress the explosion, and then use another linear feedback noise term to stabilize the system to the equilibrium point 0. Since it is almost impossible to get an closed-form solutions, the discrete-time approximation algorithm is constructed. The interpolated sequence of the discrete-time algorithm is proved to converge to the switching diffusion process, and then the regularity and stability results of the approximating sequence are derived. In Chapter 4, we study the nonlinear stochastic systems with switching. Use the similar methods, we can prove that well designed noise type feedback control could also regularize and stabilize nonlinear switching diffusions. Examples are used to demonstrate the results.

#### Recommended Citation

Zhao, Guangliang, "Properties Of Nonlinear Randomly Switching Dynamic Systems: Mean-Field Models And Feedback Controls For Stabilization" (2014). *Wayne State University Dissertations*. 946.

http://digitalcommons.wayne.edu/oa_dissertations/946