Title: Variation Evolving for Optimal Control Computation, A Compact Way,
Abstract: A compact version of the Variation Evolving Method (VEM) is developed for the
optimal control computation. It follows the idea that originates from the
continuous-time dynamics stability theory in the control field. The optimal
solution is analogized to the equilibrium point of a dynamic system and is
anticipated to be obtained in an asymptotically evolving way. With the
introduction of a virtual dimension, the variation time, the Evolution Partial
Differential Equation (EPDE), which describes the variation motion towards the
optimal solution, is deduced from the Optimal Control Problem (OCP), and the
equivalent optimality conditions with no employment of costates are
established. In particular, it is found that theoretically the analytic
feedback optimal control law does not exist for general OCPs because the
optimal control is related to the future state. Since the derived EPDE is
suitable to be solved with the semi-discrete method in the field of PDE
numerical calculation, the resulting Initial-value Problems (IVPs) may be
solved with mature Ordinary Differential Equation (ODE) numerical integration
methods.