Intermittent Stochastic Control Theory For Hybrid Systems

In recent years,hybrid stochastic systems(stochastic system with Markovian Switching)have attracted lots of attention from scholars.In the research of hybrid stochastic systems,stability and stabilization have always been one of the most popular topics,and there are abundant research results.In order to satisfy the requirements of engineering or cost saving,Mao introduced the feedback control based on discrete-time state observation into stochastic systems and studied hybrid stochastic systems discrete-time feedback stabilization problem in 2013.In 2018,Zhang et al.first used intermittent stochastic noise to stabilize nonlinear differential equations and proposed a class of intermittent stochastic perturbation stabilization theory.Nowadays intermittent control has been applied to many fields,such as applications of coupled neural network,complex network,multi-agent system.On this basis,this dissertation is a study of intermittent noise stabilization of hybrid stochastic systems and intermittent feedback control problems based on discrete-time state observation,and obtained several sufficiency criterions concerning the stability of equilibrium points and the stability in distribution.Firstly,we designed the intermittent stochastic noise as a controller to control the unstable hybrid systems.Considering the time delay unavoidable in many real systems,we proposed to add a time lag to this controller and use intermittent control to stabilize hybrid differential equations,which not only conforms to the actual situation,but also saves the control cost.However,intermittent control brings new challenges and complicates relevant theoretical analysis.Therefore,we obtain a sufficient stability criterion through M matrix,It(?) integral,comparison theorem,intermittent control strategy and other relevant techniques,and calculate the upper bound of noise rest rate and the time lag of control signal.Secondly,we investigated the intermittent stochastic noise stabilization problem based on continuoustime and discrete-time state observations for the unstable hybrid Hopfield neural networks,respectively.On the one hand,the intermittent stochastic noise stabilization based on continuous-time state observations is established by using the exponential martingale inequality and the ergodicity of Markovian chain,and the sufficient criterion for stability of hybrid stochastic neural networks is proposed.On the other hand,M matrix theory and generalized It(?) integral are used to demonstrate that the p-moment exponents stability of hybrid stochastic neural networks based on discrete-time state observations,and to estimate the lower bound of intermittent control operating time and the upper bound of time interval between two discrete-time state observations.Thirdly,for unstable hybrid stochastic system,we consider adding intermittent controllers to the drift term and using discrete-time state observations control with time lags to stabilize the hybrid stochastic systems based on the previous discussion.Using Lyapunov functional method,M matrix,generalized It(?) formula and other analytical techniques,we obtain some sufficient conditions for exponential stability.In addition,by applying the comparison theorem,the p-moment exponential stability and almost sure exponential stability of the controlled system,we obtain the upper bound of the time delay,the lower bound on the intermittent control operating time,and the upper bound of the time interval between two discrete-time state observations.Finally,in the above study,the stability is emphasized in the sense of asymptotic stability of trivial solutions,that is,the solution of the controlled hybrid system converges to an equilibrium state,but in the real system,there is no equilibrium state,so it is also of great significance to study whether the solution is stable in the distributional sense.A periodic intermittent controller with continuous-time state observation is proposed to make a given unstable hybrid stochastic system stability in distribution.On the one hand,we established the sufficient conditions for the stability in distribution of hybrid stochastic differential equations by using M matrix theory and intermittent control strategy.On the other hand,the existence and uniqueness of probability measure are proved,and the lower bound of intermittent control working time is calculated.In this paper,intermittent feedback control is used to stabilize hybrid stochastic systems,and stochastic analysis and control theory methods are used to explore the equilibrium stability and stability in distribution of hybrid stochastic systems.These theories have potential application background in many interdisciplinary fields,and have certain practical significance.