Hybrid stochastic differential equations (SDEs) (also known as SDEs with Markovian switching) have been used to model many practical systems where they may experience abrupt changes in their structure and parameters. One of the important issues in the study of hybrid systems is the automatic control, with consequent emphasis being placed on the asymptotic analysis of stability. One classical topic in this field is the problem of stabilization. The stability of hybrid systems by feedback control based on continuous-time observations has been studied extensively in the past decades. Recently, Mao  initiates the study on the mean-square exponential stabilization of continuous-time hybrid stochastic differential equations by feedback controls based on discrete-time state observations. Mao  also obtains an upperbound on the duration between two consecutive state observations. However, it is due to the general technique used there that the bound on is not very sharp. In this thesis, we will consider a couple of important classes of hybrid SDEs. Making full use of their special features, we will be able to establish a better bound on. Our new theory enables us to observe the system state less frequently (so costless) but still to be able to design the feedback control based on the discrete-timestate observations to stabilize the given hybrid SDEs in the sense of mean-square exponential stability. Moreover, we will be able to establish a better bound on making use of Lyapunov functionals. By this method, we will not only discuss the stabilization in the sense of exponential stability but also in other sense of H1 stability or asymptotic stability as well. We will not only consider the mean square stability but also the almost sure stability. It is easy to observe that the feedback control there still depends on the continuous-time observations of the mode. However, it usually costs to identify the current mode of the system in practice.So we can further improve the control to reduce the control cost by identifying the mode at discrete times when we make observations for the state. Therefore, we will also design such a type of feedback control based on the discrete-time observations of both state and mode to stabilize the given hybrid stochastic differential equations (SDEs) in the sense of mean-square exponential stability in this thesis. Similarly, we can extend our discussion to the stabilization of continuous-time hybrid stochastic differential equations by feedback controls based on not only discrete-time state observations but also discrete-time mode observations by Lyapunov method. At last, we will investigate stability of Stochastic differential delay equations with Markovian switching by feedback control based on discrete-time State and mode observations by using Lyapunov functional. Hence, we will get the upperbound on the duration between two consecutive state and mode observations.
|Date of Award||26 Sep 2018|
- University Of Strathclyde
|Supervisor||Xuerong Mao (Supervisor) & Wei Yang (Supervisor)|