Some remarks on homogenization and exact boundary controllability for the one-dimensional wave equation

SynopsisA simple model of a vibrating multidimensional structure made of a n-dimensional body and a one-dimensional straight string is introduced. In both regions (n-dimensional body and a onedimensional string) the state is assumed to satisfy the wave equation. Simple boundary conditions are introduced at the junction. These conditions, in the absence of control, ensure conservation of the total energy of the system and imply some rigidity of the boundary of the n-d body on a neighbourhood of the junction. The exact boundary controllability of the system is proved by means of a Dirichlet control supported on a subset of the boundary of the n-d domain which excludes the junction region. Some extensions are discussed at the end of the paper.

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Exact boundary controllability of coupled hyperbolic equations

International Journal of Applied Mathematics and Computer Science ◽

10.2478/amcs-2013-0052 ◽

2013 ◽

Vol 23 (4) ◽

pp. 701-709 ◽

Cited By ~ 4

Author(s):

Sergei Avdonin ◽

Abdon Choque Rivero ◽

Luz De Teresa

Keyword(s):

Wave Equations ◽

Hyperbolic Equations ◽

Basis Property ◽

Distributed Parameter ◽

Exact Boundary Controllability ◽

Boundary Controllability ◽

One Dimensional ◽

Riesz Basis Property ◽

Exact Boundary ◽

Dimensional Wave

Abstract We study the exact boundary controllability of two coupled one dimensional wave equations with a control acting only in one equation. The problem is transformed into a moment problem. This framework has been used in control theory of distributed parameter systems since the classical works of A.G. Butkovsky, H.O. Fattorini and D.L. Russell in the late 1960s to the early 1970s. We use recent results on the Riesz basis property of exponential divided differences.

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Thermally corrected solutions of the one-dimensional wave equation for the laser-induced ultrasound

Journal of Applied Physics ◽

10.1063/5.0050895 ◽

2021 ◽

Vol 130 (2) ◽

pp. 025104

Author(s):

Misael Ruiz-Veloz ◽

Geminiano Martínez-Ponce ◽

Rafael I. Fernández-Ayala ◽

Rigoberto Castro-Beltrán ◽

Luis Polo-Parada ◽

...

Keyword(s):

Wave Equation ◽

One Dimensional ◽

The One ◽

Dimensional Wave

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Сlassical solution of the mixed problem for the one-dimensional wave equation with the nonsmooth second initial condition

Doklady of the National Academy of Sciences of Belarus ◽

10.29235/1561-8323-2020-64-6-657-662 ◽

2020 ◽

Vol 64 (6) ◽

pp. 657-662

Author(s):

V. I. Korzyuk ◽

J. V. Rudzko

Keyword(s):

Boundary Condition ◽

Wave Equation ◽

Mixed Problem ◽

Method Of Characteristics ◽

Smooth Boundary ◽

Analytical Form ◽

Initial Condition ◽

One Dimensional ◽

The One ◽

Dimensional Wave

In this article, we study the classical solution of the mixed problem in a quarter of a plane and a half-plane for a one-dimensional wave equation. On the bottom of the boundary, Cauchy conditions are specified, and the second of them has a discontinuity of the first kind at one point. Smooth boundary condition is set at the side boundary. The solution is built using the method of characteristics in an explicit analytical form. Uniqueness is proved and conditions are established under which a piecewise-smooth solution exists. The problem with linking conditions is considered.

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