Time-Optimal Small Movement of a Platform with Pendulums of Two Types of Equilibrium

We consider the problem of the time-optimal small displacement of a rigid body moving forward along a horizontal straight line and carrying n mathematical pendulums. At the initial moment of time, the system is at rest, with one part of the pendulums in stable equilibrium, and the other in unstable equilibrium. The system must move the given distance with vibration damping through a single control force applied to the platform and limited in magnitude; there is no friction. The pendulums can oscillate in the vertical plane without interfering with each other due to the design, but the inverted rods must not fall (i.e., pass through the lower stable vertical position). The displacement is assumed to be small to the extent that linearized equations can be used. It is shown that when the number of inverted pendulums is odd, the movement of the platform begins from reverse, but when there is an even number, it does not. For the case of two pendulums (inverted and normal), the evolution of optimal control functions with increasing range of movement is studied. The optimal control modes found for a linear system are applied to the problem of small displacement of this object with nonlinear equations.