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Title: Stabilization of stochastic differential equations by feedback controls based on discrete-time observations
Author: Dong, Ran
ISNI:       0000 0004 7967 2685
Awarding Body: University of Strathclyde
Current Institution: University of Strathclyde
Date of Award: 2019
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Traditionally, to stabilize an unstable continuous-time stochastic differential equation (SDE) by feedback control, continuous-time observations of the system state are required. This is obviously expensive and unrealistic, so recently Mao discretized the observations. This thesis is to investigate the stabilization problem of continuous-time differential equation systems by deterministic and stochastic feedback controls based on discrete-time observations. This problem includes determining the conditions for original system and controller, and calculating the upper bound of the observation interval, namely the minimum of the observation frequency. The SDEs discussed in this thesis are all in the Itô sense. The main mathematical fundamentals used are Itôs formula, Lyapunov's second method and inequalities. The problem was investigated under Lipschitz continuity and linear growth condition. Firstly, I investigated the hybrid SDEs, which is also known as stochastic differential equations with Markovian switching. Using discrete-time observations of system state and mode, we can achieve pth moment stabilization in the sense of asymptotic and exponential stability for p > 1. Our new theory expands from the second moment to pth moment and reduces the observation frequency. Secondly, I used stochastic feedback control, which is based on Brownian motion, to stabilize non-autonomous linear scalar ODEs as well as nonlinear multidimensional hybrid SDEs. Almost sure exponential stabilization is discussed. The new established theory expands the scope of applicable original unstable systems from autonomous ODEs to non-autonomous ODEs and hybrid SDEs. Thirdly, by making full use of the time-varying system property, I used the timevarying observation intervals instead of a constant as before. Non-autonomous periodic SDEs and hybrid SDEs are investigated. Many stabilities are discussed, including asymptotic and exponential stabilities in pth moment for p > 1 and almost surely. My new established theory not only reduces the observation frequencies, but also offers flexibility on the setting of observations.
Supervisor: Mao, Xuerong Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral