Optimal Finite-Time Feedback Controllers for Nonlinear Systems with Terminal Constraints

This paper presents a dynamic-programming approach for determining finite-time optimal feedback controllers for nonlinear systems with nonlinear terminal constraints. The method utilizes a polynomial series expansion of the cost-to-go function with time-dependent gains that operate on the state variables and constraint Lagrange multipliers. These gains are computed from backward integration of differential equations with terminal boundary conditions, derived from the constraint specifications. The differential equations for the gains are independent of the states. The Lagrange multipliers at any particular time are evaluated from the knowledge of the current state and the gain values. Several numerical examples are considered to demonstrate the applications of this methodology. The accuracy of the method is ascertained by comparing the results with those obtained by using open-loop solutions to the respective problems. Finally, results of the application of the developed methodology to a spacecraft detumbling problem are presented.