  The compact Variation Evolving Method (VEM) that originates from the
continuous-time dynamics stability theory seeks the optimal solutions with
variation evolution principle. It is further developed to be more flexible in
solving the Optimal Control Problems (OCPs), by relaxing the definite
conditions from a feasible solution to an arbitrary one for the derived
Evolution Partial Differential Equation (EPDE). To guarantee the validity, an
unconstrained Lyapunov functional that has the same minimum as the original OCP
is constructed, and it ensures the evolution towards the optimal solution from
infeasible solutions. With the semi-discrete method, the EPDE is transformed to
the finite-dimensional Initial-value Problem (IVP), and then solved with common
Ordinary Differential Equation (ODE) numerical integration methods.
Illustrative examples are presented to show the effectiveness of the proposed
method.
