A Control Problem With Inconsistent Time

In the paper,we consider time inconsistency as the limit of time-consistent prob-lems and adopt a different approach from Professor Jiongmin。Yong’s to considered the existence of feedback equilibrium control for the time-inconsistent LQ problem and time-inconsistent problems by ordinary differential equations systematically.For the time-inconsistent LQ problem,controlled system is X(s)=AZ(s)+Bu(s),s∈(t,T]X(t)=x,t∈[0,T),where A,B are constant matrixs,for any initial(t,x)and u,(11)admits a unique solution Xt,xu,(·)X(·t,x,u).The objective functional is(?)Problem:for any ε>0,u ∈Rm and any initial(t,x)∈[0,T)×Rn,we find a control u∈U[t,T]such that(?)where(?)Under appropriate conditionss,we consider first that the coupled differential Riccati c-quation(?)has a semi-positive solution P,then feedback equilibrium control is given by u(t,x)=-M1(t)BTP(t)x,(?)(t,x)∈=[0,T]×RnFor the time-inconsistent problem by ordinary differential equations.controlled sys-tem is X(s)=f(s,X(s))+Bu(s),s∈(t,T],X(t)=x,Objective functional is(?)We introduce partial differential operators Dv that is Dvh(t,x)=ht(t,x)+<hx(t,x),∫(t,x)+Bv>Under the appropriate conditions,using the idea of studying the time-inconsistent LQ problem,we consider first that the coupled differential HJB equation(?)has solution(V,u),and then obtain the existence of feedback equilibrium control in the sense of(13),the problem has a feedback equilibrium feedback control uIn this paper,definition of feedback equilibrium control for time-inconsistent prob-em is introduced by us and many economists and financiers are interested in,which is different from Professor diongmin yong’s definition about the closed-loop control for time-inconsisteney.