Optimal LQ Control And Stabilization For Stochastic Systems With Multiplicative Noise

The study on stochastic control for multiplicative noise systems is one of key research areas in control theory,which has broad applications in radio communi-cation,networked control systems,economics,aerospace,etc.However,compared with additive noise systems,the research on stochastic control for multiplicative noise systems is more challenging,and therefore results in slower progress.Some relevant issues like the output feedback control and stabilization problems for mul-tiplicative noise systems have been reported as open problems.Moreover,relevant problems induced from control problems for standard multiplicative noise systems are still unsolved.For example,the optimal control and stabilization problems for mean-field systems,and the time-inconsistent stochastic control problem.This paper mainly focuses on the research on optimal LQ control and stabilization for several types of multiplicative noise systems.The main contributions are as:For the first time,the optimal output feedback control is designed for multiplicative noise system with intermittent observations,and the "separation principle" is thus verified;Moreover,the necessary and sufficient stabilization conditions are derived for multiplicative noise system with intermittent observations;For both the state packet loss case and the UDP case of networked control systems,the necessary and sufficient stabilization conditions are derived,and the maximal packet losses rate is developed for the scalar-valued case;For both discrete-time and continuous-time mean-field systems,the necessary and sufficient solvability conditions are derived for the finite horizon mean-field LQ control prob-lem;For infinite horizon case,if the weighting matrices are positive semi-definite and the exact detectable(or exact observable)assumption is satisfied,the necessary and sufficient stabilization conditions for mean-field systems are firstly derived;For general discrete-time time-inconsistent stochastic LQ control problem,the necessary and sufficient solvability conditions are developed.The main innovations are as:For multiplicative noise systems with intermit-tent observations,the barrier of "separation principle doesn't hold for multiplicative noise systems" is firstly overcame in this paper,relevant results are spread to solve the stabilization problems for NCSs of state packet loss case and the UDP case;For mean-field optimal LQ control problem,the explicit solution to forward-backward stochastic difference/differential equation is obtained for the first time,and the re-lationship between the system state and the costate is developed,the necessary and sufficient solvability conditions are thus derived;Under the basic assumptions of ex-act detectability or exact observability,by using matrix decomposition skills,the sta-bilization conditions are firstly developed,which extends the traditional stabilization results for LQ control to the weighting matrices being positive semi-definite case;It is first that the time-inconsistent LQ control problem is investigated by decoupling the forward and backward difference equationThe main contents,main results and innovations of this paper are listed as below in the order of chapters:1.The output feedback control and stabilization problems for multiplicative noise systems with intermittent observations is investigated.? Based on the intermittent observation data,the optimality of the iterative esti-mator is verified through precise deduction.By using dynamic programming approach,the explicit optimal output feedback control is derived for the first time.While for the infinite horizon stabilization problem,the Lyapunov func-tion candidate is defined with the optimal cost function,the necessary and sufficient stabilization conditions are thus derived.? For two cases in networked control systems:state packet loss case and UDP case,the obtained results are applied to solve the output feedback control and stabilization problems.Moreover,the maximal packet losses rate is developed for the scalar-valued case.The main innovation is that we overcome the barrier of "separation principle doesn't hold for multiplicative noise system",which forms the basis in solving the output feedback control and stabilization problems for multiplicative noise systems with general observations.2.We consider the optimal LQ control and stabilization problems for linear discrete-time mean-field multiplicative noise systems.Firstly,by using the convex variational methods,the maximum principle for mean-field control is derived.Next,by decoupling the forward and backward stochastic difference equation,the relation-ship between the state and the costate is developed.Thus the solvability conditions for mean-field LQ control problem are obtained.For infinite horizon mean-field control and stabilization problem,with the weighting matrices being positive semi-definite,two main results are derived:one is based on the exact detectability as-sumption,we have shown the mean-field system is mean square stabilizable if and only if the given coupled Riccati equation admits a unique positive semi-definite solution;The other one is based on the exact observability assumption,it is shown the mean-field system is mean square stabilizable if and only if the given coupled Riccati equation has a unique positive definite solution.The main innovations are that the explicit solvability conditions are obtained,which are easily verified;we extend the stabilization results for traditional LQ control to be weighting matrices being positive semi-definite case.3.The continuous-time mean-field optimal LQ control and stabilization prob-lems are solved thoroughly.Similar to the discrete-time case,the maximum princi-ple of continuous-time mean-field optimal LQ control problem is derived by using convex variational method.Moreover,the optimal control is obtained,and there-fore the solvability conditions of finite horizon LQ control are developed.For the infinite horizon optimal control and stabilization problems,by applying the matrix decomposition and assuming the weighting matrices to be positive semi-definite,we have shown that under the assumption of exact detectability(resp.exact observ-ability),the mean-field system is mean square stabilizable if and only if the given coupled ARE admits a unique positive semi-definite solution(resp.positive definite solution).It should be noted that the continuous-time mean-field LQ control and sta-bilization problems are thoroughly solved in this paper,and which forms the basis in solving the indefinite mean-field LQ control problem,portfolio selection problem and time-inconsistent LQ control problem,etc.4.The time-inconsistent stochastic LQ control problem for discrete-time sys-tem is solved in the game theoretic formulation.First,the definition of "equilibri-um control" is presented,and the maximum principle for time-inconsistent control problem is derived via variational methods.Then for the case of system state being scalar valued,by decoupling a flow of forward and backward stochastic difference equations,the relationship between the state and the costate is developed.Thus the "equilibrium control" is obtained,which is based on the nonsymmetric Riccati equations.Moreover,the necessary and sufficient solvability conditions are present-ed.The main innovations are that the general research methods are given in tackling with the time-inconsistent equilibrium control problem:First,to develop the max-imum principle,then to derive the solution to the forward and backward stochastic difference equations,finally to get the explicit "equilibrium control".It is noted the obtained results can be used to solve the mean-variance portfolio selection problem.