  A compact version of the variation evolving method (VEM) is developed in the
primal variable space for optimal control computation. Following the idea that
originates from the Lyapunov continuous-time dynamics stability theory in the
control field, the optimal solution is analogized to the stable equilibrium
point of a dynamic system and obtained asymptotically through the variation
motion. With the introduction of a virtual dimension, namely the variation
time, the evolution partial differential equation (EPDE), which seeks the
optimal solution with a theoretical guarantee, is developed for the optimal
control problem (OCP) with free terminal states, and the equivalent optimality
conditions with no employment of costates are established in the primal space.
These conditions show that the optimal feedback control law is generally not
analytically available because the optimal control is related to the future
states. Since the derived EPDE is suitable to be computed with the
semi-discrete method in the field of PDE numerical calculation, the optimal
solution may be obtained by solving the resulting finite-dimensional
initial-value problem (IVP).
