A methodology for the design of finite-horizon feedback control strategies is presented, with application to the cart-pendulum system. The control objective is to drive the system from an arbitrary initial state to the desired final state while minimizing the square of the control force. The optimal feedforward control for the linearized model is first derived. It is then converted into a feedback form to improve robustness to uncertainties. To simplify computations during the motion, a linearized feedback controller is introduced. To prevent the increase of the control force near the end of the control interval, an infinite-horizon feedback controller is applied. The effectiveness of the proposed control strategy is demonstrated through numerical experiments in the presence of dynamic and parametric uncertainties. Quasi-optimal trajectories and bounded control effort are achieved, offering a robust and practical solution for finite-horizon control problems.

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.