Feedback Control On Hybrid Stochastic Differential Equations And Its Application

The growth of science and technology and the rapid of climatic change,has resulted into the emergence of uncertainties to various systems such as ecology,finance,signal processing,epi-demiology,telecommunication,only mentioning the few.Such systems may undergo abrupt changes in their structures.To control such uncertainties,stochastic differential equations(SDEs)with Markovian switching also known as switching diffusion system(SDSs)including both continuous dynamics and discrete events,have been widely used,where by the important emphasis being placed on automatic control.For the sake of saving time and costs the feedback control based on discrete-time observations is used to stabilize the switching diffusion systems.Response lags are required by most of physical systems and play a key role in the feedback control.One of the aim of this work is to design delay feedback control functions based on the discrete-time observations in order for the controlled switching diffusion system(CSDS)to be stable and bounded.Chapter 2 considers the case of quasi-linear CSDS.For the quasi-linear CSDSs,we designed the delay feedback control functions based on the discrete-time observations of the system states and the Markovian states.We give sharp criteria on the uniform boundedness of the solution in infinite horizon as well as exponential stability in mean square.That is,by a feedback control satisfying a proposed condition,the solution will be uniformly bounded or exponentially stable in mean square,while it will be unbounded or unstable under a slightly weaker control.The main techniques are the strong ergodicity theory of Markov chains and the asymptotic analysis techniques of stochastic functional differential equations(SFDEs),which are modified from the previous works to suit with our problem.Chapter 3 deals with the case of nonlinear DCSDSs.We give the moment boundedness of the global solutions firstly.We go a further step to use the delay feedback control based on the discrete-time observations of the system states and the Markovian states to stabilize the unstable DSDs.We obtained criteria for exponential stability in pth moment,in probability one and in H_∞.Also,we revealed that,the values taken by the response lag and successive observation lag have close relationships with the estimate of the sample and moment Lyapunov exponents.The main techniques are the stochastic Lyapunov functional analysis and stochastic asymptotic analysis.Chapter 4 focuses on the application of stochastic differential equations(SDEs),where by we studied the stochastic logistic population system under regime switching.We designed the delay feedback control based on the discrete-time observations and give the criteria for permanence and extinction of the stochastic population system in the time average and in almost surely.We estimated the boundary of the time delay and the observation delay.The main techniques applied are stochastic comparison theorem and stochastic asymptotic analysis.Chapter 5 gives the conclusion and further recommendations.This dissertation is categorised into five sections(Chapters).Chapter 1 tells about the general knowledge which include the background of the feedback control and the specific knowledge on SDE which have been applied in this dissertation.Chapters 2,3 and 4 are our findings.Chapter5 gives the concluding remarks of this dissertation.