Optimal Movement of a Platform with an Inverted Pendulum

We consider the problem of the optimal horizontal displacement of a platform with a pendulum over the greatest distance in the given time by means of a limited force applied to the platform. At the initial and final moments of time, the system is at rest, and in both states the pendulum is in the upper (unstable) equilibrium position. During the movement, it can pass through the lower position. There is no friction and kinematic restrictions are not imposed on the system (constructively excluded). Trajectories satisfying the necessary optimality conditions are constructed numerically. The evolution of the optimal control functions with a change in the assigned movement time is shown in a special diagram.