IMPLEMENTATION OF THE APPROACH OF OSCILLATORY SYSTEMS BASED ON THE PRINCIPLE OF TIME DILATION

Abstract

The paper considers the problem of the approach of two controlled systems describing the dynamics of mathematical pendulums, in which one of the objects seeks to achieve thе meeting, and the other to avoid it. In order to apply the first direct method of L.S. Pontryagin, to solve the problem, a modification of this method was required, based on the application of the time dilation principle. The reason is that the Pontryagin condition, which is the basis of the first direct method and, in fact, provides the possibility of constructing the control at each instant of time according to the current control of the evader, is not satisfied for the problem at hand. This condition reflects the advantage of the pursuer over the evading object in control resources, expressed through the parameters of the systems. A modification of the Pontryagin condition is used, which includes the so-called time dilation function, which plays a decisive role in the construction of the control of the pursuer on the basis of the evaderʼs control in the past, as it were, on the basis of delayed information. For the problem under study, an appropriate function of time dilation is introduced and conditions are derived that ensure the possibility of meeting of the objects in a prescribed finite time. Also, formulas are given that describe the way of constructing the pursuer control on the basis of the adversary control in the past. Using software, a visual illustration of the process of convergence of the objects on the plane, provided the evader is moving in a stable orbit, is created. The algorithm for constructing the current control of the pursuer that leads to the meeting is described.