Analysis of minimum time before stabilization of oscillatory process under different modes

The paper presents a numerical approach to solving the problems of optimal control over establishing oscillatory systems by means of boundary controls. In contrast to the reviewed works of other authors on similar topics, the problem of establishing the oscillatory process is numerically studied in a temporal neighborhood of the required mode, the friction force being taken into account in the mathematical model of the oscillatory system. The schemes of deriving the basic formulas are presented, as well as algorithms for their numerical solution by first-order optimization methods. The corresponding analytical formulas for the components of the gradient of the target functional for control actions are obtained. A comparison is made of the minimum time for the oscillation process to be established both with its optimal control and with the process self-determination. The results of numerical experiments are presented. The results can be attributed to the different processes described by hyperbolic equations.