Exact Controllability Of Linear Mean-Field Stochastic Systems And Observability Inequality For Mean-Field Bsdes

This paper is concerned with the controllability of the linear mean-field stochastic system with time-varying random coefficients.we consider the exact controllability the controlled equation we give the dual equation of the system By the existence and uniqueness of solutions of the mean-field backward s-tochastic differential equations,we know that there exists a unique solution for the above equation.By applying Ito formula to(x(·),y(·))on the interval[0,T],we have the following duality relation:where(?)if exist a const C>0,satisfy?yT?LFT2?C???LF2,2,We call it the observability inequality of the dual equation.We use the observ-ability inequality as an important tool to deal with the exact controllability problem.Finally,we obtain the equivalence relation between L2-exact con-trollability and L2-observability inequality.In order to obtain the equivalence relation,we introduce a family of opti-mal control problems.Define a functionalJ(.;x0,xT):LFT2(?;Rn)?R,(?)If we make x0?Rn,xT ?LFT2(?;Rn)as parameter,yT?LFT2(?;Rn)as control,the dual equation as status,minimize the function J form a family of optimal control problem.We prove that the L2-exact controllability of the con-trolled system is equivalent to the L2-observability inequality of the dual equa-tion and equivalent to optimal control problem exist unique optimal control.As an application of this result for any x0? Rn,any xT? LFT2(?;Rn),over the L2-feasible control set u(x0,xT):= {u?LF2(?;L2(0,T;Rn))|x(T;x0,u)=xT}minimize ?u?LF2,2.We use the previous results to give the optimal control of the norm optimal control problem.The paper is organized as follows:The first chapter is about the back-ground of the mean-field equations and the history of exact controllability.In the second chapter,we list some notations,preliminaries and some result-s about mean-field stochastic differential equations and mean-field backward stochastic differential equations.The third chapter is about the main results.As the proof of the problem is long,the proof is divided into several parts that are easier to understand by several lemmas.In the fourth chapter,we apply our result to a norm optimal control problem.The fifth chapter is the prospect,mainly introduced some work we could do in the future.