  The Variation Evolving Method (VEM) that originates from the continuous-time
dynamics stability theory seeks the optimal solutions with variation evolution
principle. After establishing the first and the second evolution equations
within its frame, the third evolution equation is developed. This equation only
solves the control variables along the variation time to get the optimal
solution, and its definite conditions may be arbitrary since the equation can
eliminate possible infeasibilities. With this equation, the dimension of the
resulting Initial-value Problem (IVP), transformed via the semi-discrete
method, is greatly reduced. Therefore it might relieve the computation burden
in seeking solutions. Illustrative examples are solved and it is shown that the
proposed equation may produce more precise numerical solutions than the second
evolution equation, and its computation time may be shorter for the dense
discretization.
