  An effective form of the Variation Evolving Method (VEM), which originates
from the continuous-time dynamics stability theory, is developed for the
classic time-optimal control problem with control constraint. Within the
mathematic derivation, the Pontryagin's Minimum Principle (PMP) optimality
conditions are used. Techniques including limited integrator and corner points
are introduced to capture the right solution. The variation dynamic evolving
equation may be reformulated as the Partial Differential Equation (PDE), and
then discretized as finite-dimensional Initial-value Problem (IVP) to be solved
with common Ordinary Differential Equation (ODE) integration methods. An
illustrative example is solved to show the effectiveness of the method. In
particular, the VEM is further developed to be more flexible in treating the
boundary conditions of the Optimal Control Problem (OCP), by initializing the
transformed IVP with arbitrary initial values of variables.
