ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

# Differential Equations and Control Processes (Differencialnie Uravnenia i Protsesy Upravlenia)

## Optimal Boundary Control of Oscillation Process by Displacement at Two Ends of a Rod Consisting of Two Sections of Different Density and Elasticity

### Author(s):

V. R. Barseghyan

Institute of Mechanics of National Academy of Sciences of RA
Yerevan State University

barseghyan@sci.am

### Abstract:

We consider the problem of optimal boundary control for a one-dimensional wave equation, describing the longitudinal oscillations of a heterogeneous rod consisting of two heterogeneous sections or transverse vibrations of a heterogeneous string with given initial and final conditions. In this case, it is assumed that the time of propagation of the wave along each of the sections is the same. The control is carried out by displacement at the two ends. The quality criterion is given over the entire time period. A constructive approach is proposed for constructing an optimal boundary control, which is carried out according to the following scheme. The problem is reduced to the problem of control of distributed actions with zero boundary conditions, then the method of separation of variables and the methods of the theory of optimal control of finite-dimensional systems are used. The obtained results are illustrated by a specific example.

### Keywords

• longitudinal oscillations of a piecewise homogeneous rod
• optimal boundary control
• optimal control of oscillations
• separation of variables
• transverse vibrations of a piecewise homogeneous string

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