#### Access Type

Open Access Dissertation

#### Date of Award

January 2013

#### Degree Type

Dissertation

#### Degree Name

Ph.D.

#### Department

Mathematics

#### First Advisor

Prof. Gang G. Yin

#### Abstract

This dissertation is focuses on near-optimal controls for stochastic differential equation with regime switching. The random switching is presented by a continuous-time Markov chain. We use the idea of relaxed control and mean of martingale formulation to show a weak convergence result.

The first chapter is devoted to the study of stochastic Li´enard equations with random switching. The motivation of our study stems from modeling of complex systems in which both continuous dynamics and discrete events are present. The continuous component is a solution of a stochastic Li´enard equation and the discrete component is a Markov chain with a finite state space that is large. A distinct feature is that the processes under consideration are time inhomogeneous. Based on the idea of nearly decomposability and aggregation, the state space of the switching process can be viewed as "nearly decomposable" into l subspaces that are connected with weak interactions among the subspaces. Using the idea of aggregation, we lump the states in each subspace into a single state. Considering the pair of process (continuous state, discrete state), under suitable conditions, we derive a weak convergence result by means of martingale problem formulation. The significance of the limit process is that it is substantially simpler than that of the original system. Thus, it can be used in the approximation and computation work to reduce the computational complexity.

Finally, we investigate the system behavior of Van der Pol oscillator by introducing the noise. The system have been performed numerically and results are shown using Matlab. Simulations show that the proposed model gives limit cycles are more accurate as the noise decreased which the limit cycle is close to a sinusoidal oscillation and the shape of the signal becomes less sinusoidal as the noise increased.

#### Recommended Citation

Talafha, Yousef, "Two-Time-Scale Systems In Continuous Time With Regime Switching And Their Applications" (2013). *Wayne State University Dissertations*. 860.

http://digitalcommons.wayne.edu/oa_dissertations/860