The report discusses several model tasks for controlling a platform with an attached mathematical pendulum or a double pendulum. Small oscillations are considered. Optimal control is the force acting horizontally on the platform, or the moment acting on the driving wheels of the platform. The driving wheel model is discussed separately. The platform is transferred in a straight line from a state of rest at a given distance to a state of rest in a given time. The minimization of energy approach is considered. Vibration damping occurs. Expressions for optimal control by the Pontryagin maximum method and using the generalized Gauss principle are found.

Several problems are considered, depending on how the pendulum is positioned and attached.
a). The initial position of the mathematical pendulum is vertically upward (unstable position of the pendulum). The pendulum is pivotally attached to the platform.
b). The balance position of the pendulum and the initial position are vertically downwards.
c). The pendulum is facing up, attached to the platform along with a spiral spring. The equilibrium position of the pendulum is deflected from the vertical by a given angle. The problem takes into account the mass of the trolley, the mass of the wheels, the mass of the motors attached to the wheels, the mass of the pendulum load, the mass of the pendulum rod.
d). A manipulator consisting of two weighty rods and a load at the end is attached to the platform. The lower rod is attached to the platform together with a spiral spring, the second rod is attached to the first one either rigidly or with a spiral spring. Unlike the previous model problems, the number of degrees of freedom increases.