The Problem of Optimal Control of String Vibrations
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International Applied Mechanics, Vol. 56, No. 4, July, 2020
THE PROBLEM OF OPTIMAL CONTROL OF STRING VIBRATIONS V. R. Barsegyan1,2
The problem of optimal control of string vibrations with given initial and final conditions, and nonseparated values of derivatives of deflection functions at intermediate moments of time with a performance criterion defined over the whole period is considered. The problem is solved using the methods of separation of variables and the theory of optimal control of finite-dimensional systems with nonseparated multipoint intermediate conditions. As an example of the proposed approach, the effect of optimal string vibration with given nonlocal values of string velocities at two intermediate points of time is constructed. Keywords: string vibration, optimal control of vibrations, intermediate moments, nonseparated multipoint conditions, optimal control Introduction. A large class of physical processes associated with vibrational systems is modeled by the wave equation [10, 11, 18]. In this case, control problems often arise when it is necessary to generate specified characteristics of vibrations that satisfy intermediate conditions. The attention of researchers was attracted by multipoint boundary-value control problems, in which, along with the classical boundary (initial and final) conditions, nonseparated (nonlocal) multipoint intermediate conditions are also specified [1–9, 13, 14, 16]. Unseparated multipoint boundary-value problems arise, on the one hand, as mathematical models of real processes; on the other hand, for many equations it is impossible to correctly formulate local boundary-value problems. The inseparability of multipoint conditions is also caused by the impossibility to measure the parameters of the state of an object instantly or at its individual points. The problems of control and optimal control of vibrational processes, both external and boundary control effects for various types of boundary conditions, are considered in [1, 3, 5–7, 9, 13 14, 16], where various methods of solution are proposed. In this paper, we consider the optimal control problem for the string vibration equation with given initial conditions and nonseparated values of the velocities of the string points at intermediate times with a performance criterion set for the entire time interval. The method of separation of variables reduces the problem to an optimal control problem with a countable number of ordinary differential equations with given initial, final, and nonseparated multipoint intermediate conditions. Using the methods of the theory of optimal control of finite-dimensional systems with multipoint intermediate conditions, the optimal control effect is found. As an application of the proposed constructive approach, an optimal control effect for the vibration of a string with given nonseparated values of the velocities of the points of the string at two intermediate times is found. 1. Problem Statement. Consider a stretched homogeneous elastic string of length l with fixed ends. Let distrib
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