Variation evolving for optimal control computation in the primal space

The computation of optimal control problems (OCPs) is numerically challenging. Inspired by the Lyapunov continuous-time dynamics stability theory, a contracted variation evolving method that is easy-to-use is developed in the primal variable space, and the optimal solutions are sought asymptotically as the stable equilibria of dynamic systems through the variation motion. With the introduction of a virtual variation time, the evolution partial differential equation (EPDE) that theoretically guarantees the convergence to the optimal solution is developed for the OCP with free terminal states, and the equivalent costate-free optimality conditions are established. Since the derived EPDE is suitable to be computed with the semidiscrete method in the field of PDE numerical calculation, the optimal solution may be obtained efficiently by solving the resulting finite-dimensional initial-value problem (IVP) on ordinary differential equations.