On a problem of controlling a moving cart with elastic rod

This article discusses the problem of controlling the process of longitudinal oscillations of an elastic homogeneous rod of uniform cross section. A rod is understood as a body, the length of which significantly exceeds its cross dimensions. The rod is on a moving cart, the right end of which is rigidly fixed, and the left end is not fixed. There is no friction between the rod and the cart surface in the problem under consideration. When the cart moves, the rod performs constrained longitudinal oscillations in a non-inertial frame of reference associated with the cart. The control is the acceleration of the cart, the magnitude of which is limited. The boundaries of its accepted values are set. The value of the combined external forces acting on the rod is not known exactly, but only its limits of variation are given. The purpose of the control process is to ensure that at a given moment in time, the average value of the stretch of the rod is within a given interval. This average is calculated using the specified function.
In order to solve the problem, the method of optimizing a guaranteed result is applied. A transition to a new one-dimensional variable is made, with the help of which the considered problem of control of the longitudinal oscillations of a rod is reduced to a similar control problem in the presence of noise. The necessary and sufficient conditions are found, under which it is possible to accomplish the set goal for any admissible external forces, the total value of which satisfies the given constraints. A corresponding algorithm for constructing the law of variation of the cart acceleration is proposed. An example that clearly shows how to build the cart control, which will guarantee the achievement of the set goal, has been analyzed.