Near optimal finite-time terminal controllers for space trajectories via SDRE-based approach using dynamic programming

This paper presents a novel development to synthesize finite-time near optimal feedback control for nonlinear systems with nonlinear terminal constraints such as hypersurfaces. Especially when terminal hypersurfaces are posed as transcendental equations, the developed SDRE-based method contributes first-ever treatment for such cases. The SDRE-based approach, to synthesize continuous-time terminal controllers, is first extended for the fixed-final-time optimal control problems via solving the pointwise governing Hamilton-Jacobi-Bellman equation subject to the pseudo-linear dynamical system with linear terminal constraints. Then, to fit these derived settings into a general class of terminal constraints as hypersurfaces, the method of successive linearization is employed to obtain approximated hyperplane which facilitates state-dependent boundary conditions in order to compute the feedback control input. To establish the developed methodology, numerical investigations on nonlinear systems including the fixed-finite-time optimal control problem of spacecraft spin maneuvers with a variety of terminal cases are illustrated with details. The obtained feedback solution, for all the examples, is compared with the respective openloop solution to validate the efficacy of the novel approach that accomplishes a very high accuracy of the synthesized terminal controller incurring the least cost-to-go even though terminal hypersurface has multiple endpoint solutions which are not a priori known.