Local existence of open-loop equilibrium controls for time-inconsistent deterministic linear-quadratic problems

This paper investigates open-loop equilibrium controls for time-inconsistent deterministic Linear-Quadratic (TIDLQ for short) problems, where time inconsistency stems from non-exponential discounting in the cost functional. In contrast to most existing studies, which typically characterize open-loop equilibria through non-symmetric Riccati-type equations, we develop a symmetric Riccati framework using a refined variational approach. By examining the solvability of the symmetric Riccati equation over a restricted time interval, we establish the local existence of a linear open-loop equilibrium in the state variable.