A lot of practical systems, whose structure and parameters may change abruptly, can be modeled by the stochastic differential equations driven by continuoustime Markov chain (also known as hybrid SDEs). Among many interesting topics, stability has drawn intensive attention. In the case when a given hybrid SDE is unstable, it is a general practice to use state feedback control to achieve stabilisation. Theoretically, the design of feedback control is based on continuoustime state observations. However, in practice, it will be extremely costly and impossible to have continuous observations of the state for all time. So it is more realistic and costs less if the state is only observed at discrete times.
(Mao 2013) started the study of discretestatefeedback stabilisation of hybrid SDEs. After several years’ development, most results paid much attention to hybrid SDEs satisfying the classical linear growth condition, which could exclude many important real models. In this thesis, we will hence investigate this stabilisation problem without this restrictive condition, namely in highly nonlinear area.
We will firstly make some improvements on the existing results on this stabilisation problem of highly nonlinear hybrid SDEs. A new method will be given to estimate the difference between currenttime state and discretetime state, so that conditions imposed on the underlying systems will be less restrictive. To determine the upper bound of the observation duration, we will use optimisation method to avoid searching for free parameters and make the control rules be much more easily to verify.
Then, by taking different system structures (except for different system coefficients) in different Markovian modes into consideration, we will study the structured stabilisation of hybrid SDEs. The system structures are classified according to the view of Khasminskiitype condition. The control function will be designed in a bounded state area, rather than every observable state, in order to reduce control cost. Further, we will extend the structured stabilisation problem to hybrid stochastic delay differential equations (SDDEs). The time delays will meet a weak condition, rather than the usually seen but restrictive differential assumption. In this case, more time delays such as piecewise constant delay could be included. Moreover, time delays could influence our modestructure classification scheme.
By making use of Lyapunov functional method and integral transform (for hybrid SDDEs), H∞ stability, almost surely asymptotic stability, mean square exponential stability could be achieved. However, constructing an appropriate Lyapunov functional is always challenging, especially when integral transform method is invalid for some kinds of hybrid SDDEs, or the underlying systems are discontinuous. In this case, Razumikhin method, which is aimed to stability analysis for delay equations, will be powerful, since the discretetime state feedback control itself is also a delay segment.
We will use Razumikhin idea to the stabilisation of hybrid SDDEs, where time delays will be relatively relaxed with little restriction. In other words, we are not able to use integral transform to deal with time delays now such as discretetime delays. Next we will use this technique to study the stabilisation of hybrid SDEs by discretetime state feedback control working intermittently and having rest time. Some important stability properties will be obtained in the sense of pth moment exponential stability and almost surely exponential stability.
The successful applications to stochastic volatility model, neural networks, and oscillator systems demonstrate the practicability of our theory.
Date of Award  24 Jul 2024 

Original language  English 

Awarding Institution   University Of Strathclyde


Sponsors  University of Strathclyde 

Supervisor  Xuerong Mao (Supervisor) & Mohammud Foondun (Supervisor) 
