Vehicle trajectory reconstruction is an indispensable step before using the observed trajectory data for analysis. A recurrent challenge of the existing smoothing/filtering-based methods is the design of the smoothing parameters to avoid over-smoothing while ensuring realistic vehicle dynamics, but they often fall short in two-dimensional (2D) vehicle movements with coupled longitudinal and lateral vehicle motion. To tackle this challenge, we propose a novel approach to reconstruct vehicle trajectories based on constrained optimal control. The proposed approach outputs 2D trajectories to minimize the errors of the reconstructed trajectory with respect to the measured trajectory while respecting the vehicle dynamics and motion constraints. Bounded curvature and acceleration are used as the control variables that resemble human driver behaviour, and plausible ranges of the 2D motion variables are set as the state constraints of the optimal control problem. The proposed model is validated using both the pNEUMA trajectory dataset and a new high-precision trajectory dataset. Results show that the average Euclidean distance between the reconstructed and measured trajectory points is 0.040 m and the vehicle motion variables are all strictly within the permitted range.