Further Study On Challenging Control Problems For Stochastic Systems With Multiplicative Noises

This thesis is mainly concerned with optimal control for stochastic systems with multiplicative noise.The stabilization problem for Ito stochastic system with indef-inite control and state weighting matrices in the cost functional,the LQ control for discrete-time Markov jump linear systems involving multiplicative noise over infi-nite horizon,the LQ control for discrete-time systems involving input delay and col-ored multiplicative noise,the optimal control and stabilization problems for discrete-time networked control systems with both input delay and Markovian packet losses in the communication channel,and optimal deterministic control for discrete-time systems with multiplicative noises and random coefficients are mainly considered.The main contributions are as:First,when the state and control weighting ma-trices in the linear quadratic cost functional are allowed to be indefinite,necessary and sufficient conditions for mean-square stabilization for Ito stochastic system are given by considering the convergence of generalised differential Riccati equation;Second,different from literatures adopting complete square method,using maxi-mum principle,necessary and sufficient conditions for the solvability and mean-square stabilization for discrete-time Markov jump system with multiplicative noise are obtained;Third,for discrete-time systems involving input delay and colored mul-tiplicative noise,in terms of coupled Riccati-type difference equations,necessary and sufficient conditions for optimal control problem over finite horizon are given and the optimal controller and the optimal cost value are shown,which lays an im-portant foundation for solving more complex stochastic LQ control problem;Forth,for discrete-time networked control systems with both input delay and Markovian packet losses in the communication channel,necessary and sufficient conditions for the solvability and mean-square stabilization are given by the coupled Riccati-type difference equations with Markov chain,and note that the existing results can only deal with the LQ control problem of networked systems with Markov packet loss but delay-free or with time-delay but Bernoulli packet losses;Finally,for discrete-time system with random coefficients,using stochastic maximum principle,necessary and sufficient conditions for the solvability of LQ deterministic control are shown.The key techniques and innovations to solve the above-mentioned challenges can be attributed to solving forward-backward stochastic difference equations by de-coupling method.For non-standard LQ control problems(such as systems involving time delay,colored noise,Markov chain,etc.),it is difficult to determine the form of optimal controller in advance.In this thesis,forward-backward stochastic difference equations are derived by the stochastic maximum principle,in terms of the Riccati-type difference equation,the analytical solution of the forward-backward stochastic difference equation is given by decoupling method,then,the analytical solution of the LQ control problem is shown.This method plays a key role in solving the above-mentioned challenging problemsThe decoupling method mentioned in this paper provides an effective tool for solving other complex LQ problems.For example,the LQ control for networked systems with time-delay and packet loss adopted hold-input compensation strategy(i.e.the latest available control signal stored in the actuator buffer is used);the LQ control problem for the networked control systems with time-delay and Markovian packet loss,where multiple controllers exist in systems and each controller receives different information;the LQ deterministic control problem for Ito systems with random coefficients which are correlated with state,etc.They are more general but more complex to be dealt with.The decoupling method mentioned in this paper can be used to solve these problems.