On the Optimal Control Function Diagrams in the Problem of the Movement of a Platform with Oscillators

Abstract

We consider the problem of the time-optimal movement of a rigid body moving translationally along a horizontal straight line and carrying n linear oscillators. The only control force is applied to the platform and is limited in magnitude, and there is no friction. The system is transferred from a state of rest to a specified distance with vibration damping. The evolution of optimal control functions depending on the distance of the movement is investigated. A general approach to construct a visual diagram reflecting this evolution is proposed. To do this, a geometric interpretation of the necessary optimality conditions is used as properties of an auxiliary control curve in n-dimensional space. Numerical examples of constructing diagrams of the optimal control functions for a platform with three oscillators are given.