scholarly journals Boundary observability and exact controllability of strongly coupled wave equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Ali Wehbe ◽  
Marwa Koumaiha ◽  
Layla Toufaily
Keyword(s):  
Boundary Control ◽  
Wave Equations ◽  
Coupling Parameter ◽  
Exact Controllability ◽  
Homogeneous System ◽  
Arithmetic Property ◽  
Coupled Wave Equations ◽  
Hilbert Uniqueness Method ◽  
Boundary Observability ◽  
Dimension Space

<p style='text-indent:20px;'>In this paper, we study the exact controllability of a system of two wave equations coupled by velocities with boundary control acted on only one equation. In the first part of this paper, we consider the <inline-formula><tex-math id="M1">\begin{document}$ N $\end{document}</tex-math></inline-formula>-d case. Then, using a multiplier technique, we prove that, by observing only one component of the associated homogeneous system, one can get back a full energy of both components in the case where the waves propagate with equal speeds (<i>i.e.</i> <inline-formula><tex-math id="M2">\begin{document}$ a = 1 $\end{document}</tex-math></inline-formula> in (1)) and where the coupling parameter <inline-formula><tex-math id="M3">\begin{document}$ b $\end{document}</tex-math></inline-formula> is small enough. This leads, by the Hilbert Uniqueness Method, to the exact controllability of our system in any dimension space. It seems that the conditions <inline-formula><tex-math id="M4">\begin{document}$ a = 1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ b $\end{document}</tex-math></inline-formula> small enough are technical for the multiplier method. The natural question is then : what happens if one of the two conditions is not satisfied? This consists the aim of the second part of this paper. Indeed, we consider the exact controllability of a system of two one-dimensional wave equations coupled by velocities with a boundary control acted on only one equation. Using a spectral approach, we establish different types of observability inequalities which depend on the algebraic nature of the coupling parameter <inline-formula><tex-math id="M6">\begin{document}$ b $\end{document}</tex-math></inline-formula> and on the arithmetic property of the wave propagation speeds <inline-formula><tex-math id="M7">\begin{document}$ a $\end{document}</tex-math></inline-formula>.</p>

The theory of the limiting polarization of radio waves reflected from the ionosphere

1952 ◽  
Vol 215 (1121) ◽  
pp. 215-233 ◽  
Keyword(s):  
Electron Density ◽  
Wave Equations ◽  
Collision Frequency ◽  
Coupling Parameter ◽  
Specific Model ◽  
Refractive Indices ◽  
Radio Waves ◽  
High Frequencies ◽  
Characteristic Wave ◽  
Coupled Wave Equations

In the theory of the propagation of radio waves through a homogeneous ionized medium it is well known that ‘characteristic’ waves, sometimes called the ‘ordinary’ and ‘extra ordinary’ waves, are propagated independently. The refractive index and polarization for each characteristic wave are given by the magneto-ionic theory (Appleton 1932). If the medium is slowly varying, Booker (1936) has shown that in many cases this theory may still be applied. But there are important cases where the characteristic waves are not independent, and there is then said to be ‘coupling’ between them. This paper discusses the coupling which occurs in the lower part of the ionosphere. Here there is a ‘limiting’ region where a downcoming characteristic wave acquires the limiting polarization observed at the ground. Booker (1936) gave an approximate specification for the level of the limiting region. This paper gives a more precise specification and develops a method for calculating the limiting polarization of a downcoming characteristic wave. The theory is based on Fӧrsterling’s (1942) coupled wave equations, which apply only to vertical incidence. They contain a coupling parameter, ѱ , which depends on the gradients of electron density and collision frequency. The level of the limiting region is specified in terms of ѱ and the refractive indices of the characteristic waves. The properties of a specific model of the ionosphere are discussed, and it is shown that for frequencies greater than about 1 Mc/s the limiting polarization is that given by the magneto-ionic theory for a certain ‘limiting point’ which occurs at a definite value of the height. This value may in general be complex, but in practical cases is almost purely real and occurs where the electron density and collision frequency are small, so that at high frequencies the limiting polarization is determined only by the magnitude and direction of the earth’s magnetic field in the ionosphere.

scholarly journals A note on the exact boundary controllability for an imperfect transmission problem

2021 ◽  
Author(s):  
S. Monsurrò ◽  
A. K. Nandakumaran ◽  
C. Perugia
Keyword(s):  
Exact Controllability ◽  
Dirichlet Condition ◽  
Adjoint Problem ◽  
Imperfect Interface ◽  
External Boundary ◽  
Exact Boundary Controllability ◽  
Multipliers Method ◽  
Exact Boundary ◽  
Hilbert Uniqueness Method ◽  
Uniqueness Method

AbstractIn this note, we consider a hyperbolic system of equations in a domain made up of two components. We prescribe a homogeneous Dirichlet condition on the exterior boundary and a jump of the displacement proportional to the conormal derivatives on the interface. This last condition is the mathematical interpretation of an imperfect interface. We apply a control on the external boundary and, by means of the Hilbert Uniqueness Method, introduced by J. L. Lions, we study the related boundary exact controllability problem. The key point is to derive an observability inequality by using the so called Lagrange multipliers method, and then to construct the exact control through the solution of an adjoint problem. Eventually, we prove a lower bound for the control time which depends on the geometry of the domain, on the coefficients matrix and on the proportionality between the jump of the solution and the conormal derivatives on the interface.

Regime independent coupled-wave equations in anisotropic photorefractive media

2009 ◽  
Vol 95 (3) ◽  
pp. 589-596 ◽  
Author(s):  
K. R. Daly ◽  
G. D’Alessandro ◽  
M. Kaczmarek
Keyword(s):  
Wave Equations ◽  
Coupled Wave Equations ◽  
Coupled Wave

Topological solitons, cnoidal waves and conservation laws of coupled wave equations

2013 ◽  
Vol 87 (12) ◽  
pp. 1233-1241 ◽  
Author(s):  
E. V. Krishnan ◽  
A. H. Kara ◽  
S. Kumar ◽  
A. Biswas
Keyword(s):  
Conservation Laws ◽  
Wave Equations ◽  
Cnoidal Waves ◽  
Topological Solitons ◽  
Coupled Wave Equations ◽  
Coupled Wave
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