Synthesis of Time-Optimal Control for an Inverse Linear Pendulum on a Cart

A mechanical system that is an inverted pendulum mounted on a cart is considered. The cart moves along a horizontal line under the action of a bounded control force. The pendulum can deviate in a vertical plane passing through this line. The dynamics of the system is studied in a linear approximation. Based on Pontryagin’s Maximum Principle, the problem of synthesizing the optimal speed of movement of the cart to a given point on the line with the pendulum stopping in the upper position is solved. A description of the sets in the phase space on which the control function changes sign is given, and the behavior of the optimal trajectories is studied. The approach used consists of sequentially solving the problems of time-optimal control for subsystems of a lower dimension.