Power series solutions to the time-varying dynamic programming equations

In this paper we construct high-order approximate solutions to the value function and optimal control for a finite-horizon optimal control problem for time-varying discrete-time nonlinear systems. The method consists in expanding the dynamic programming equations (DPE) in a power series, collecting homogeneous polynomial terms and solving for the unknown coefficients from the known and previously computed data. The resulting high-order equations are linear difference equations for the unknown homogeneous terms and are solved backwards in time. The method is applied to construct high-order perturbation controllers around a nominal optimal trajectory.