Open-loop equilibrium strategy of time-inconsistent deterministic linear-quadratic problems

This paper investigates time-inconsistent linear-quadratic control problems within an open-loop framework, where the objective functional incorporates a non-exponential discounting structure. By introducing a two-point boundary value problem and a Riccati-type equation with an error function, it is established, under appropriate conditions, that the existence of an open-loop equilibrium strategy for the time-inconsistent control problem is equivalent to the existence of solutions to both the two-point boundary value problem and the Riccati-type equation. Furthermore, through an illustrative example, it is demonstrated that there exists an essential distinction between open-loop and closed-loop equilibria in time-inconsistent control problems.