Finite-Horizon Optimal Feedback Control Design Illustrated by Point Mass and Gravity Pendulum Cases

The problem of finite-horizon optimal feedback control is posed, with the goal of minimizing the functional of the square of the control force. Optimal control strategies for two simple systems, a point mass and a gravity pendulum, are derived. An algorithm for designing a feedback control strategy based on the derived optimal control is proposed. The approach is validated through numerical experiments, demonstrating the effectiveness of the optimal feedback control. The stability and quasi-optimality of the proposed control strategy is demonstrated even in the presence of uncertain parameters and unaccounted nonlinearity. The algorithm can be used for more complex systems if optimal control can be found for arbitrary initial conditions.