Equilibrium Control Of A Class Of Time-Inconsistent Differential Game

Since Issacs studied differential games in the early 1950s,the study of differential games has been in the ascendant.Especially since the time-inconsistent control problem has become a hot topic at the forefront of mathematics and finance,time inconsistent differential games have become a hot topic that has been widely concerned.This paper systematically studies a class of time-inconsistent linear quadratic zero-sum differential games and non-zero-sum differential games,in which the controlled system is the follow-ing ordinary differential equation:(?),and the performance indicator is (?), where the matrix and vector have appropriate dimensions,the discount function in the objective functional is a general non-exponential function,and the above problem is a class of widely time-inconsistent linear quadratic differential games.Time-inconsistent linear quadratic differential games are not only very important in itself,but also a break-through in the in-depth study the general theory of time-inconsistent differential games.For time-inconsistent linear quadratic zero-sum differential games and time-inconsis-tent linear quadratic non-zero-sum differential games,we introduce open loop saddle point(open loop Nash equilibrium)and closed loop saddle point(closed loop Nash e-quilibrium),as well as two-point boundary value problems and Riccati equations,re-spectively.First,we obtain the necessary and sufficient conditions that the problems with open loop saddle point(open loop Nash equilibrium)and closed loop saddle point(closed loop Nash equilibrium)are equivalent to the solvability of the two-point bound-ary value problem and the Riccati equation,respectively.Under suitable conditions,we obtain the existence and uniqueness of solutions to the Riccati equation system,and then construct a closed loop control that is proved to be the closed loop saddle point(closed loop Nash equilibrium)of the original problem.These conclusions and corresponding re-search methods undoubtedly lay a foundation for the in-depth study of time inconsistent differential games.