The Variation Evolving Method (VEM), established through the variational dynamic analysis, facilitates the computation of Optimal Control Problems (OCPs) by transforming them to Initial-value Problems (IVPs) with respect to the variational time $\tau $ . In this paper, it is reformulated in the primal variable space for better performance. The Evolution Partial Differential Equation (EPDE), which theoretically guarantees the variational motion of control $\boldsymbol{u}$ towards the optimal solution, is derived in the first-order and the second-order forms. The costate-free optimality conditions are established, and the explicit analytic expressions for the costates and the Lagrange multipliers adjoining the terminal constraints are presented, in terms of the state and the control variables. Using the semi-discrete method, the EPDE is solved as a finite-dimensional IVP on Ordinary Differential Equations (ODEs), and it is proved that with a reasonable step-size, the ODE numerical solution for the transformed nonlinear stable dynamical system is precise and has an exponential convergence rate to the optimal solution.
