Minimization of the L p -Norm, p ≥ 1 of Dirichlet-Type Boundary Controls for the 1D Wave Equation

2017 ◽  
pp. 85-94
Author(s):  
Ilya Smirnov ◽  
Anastasia Dmitrieva
Keyword(s):  
Wave Equation ◽  
Dirichlet Type ◽  
Boundary Controls ◽  
Type Boundary

scholarly journals Exact controllability for a semilinear wave equation with both interior and boundary controls

2005 ◽  
Vol 2005 (6) ◽  
pp. 619-637 ◽  
Author(s):  
Bui An Ton
Keyword(s):  
Wave Equation ◽  
Exact Controllability ◽  
Open Domain ◽  
Semilinear Wave Equation ◽  
Boundary Controls ◽  
Feedback Laws

The exact controllability of a semilinear wave equation in a bounded open domain ofRn, with controls on a part of the boundary and in the interior, is shown. Feedback laws are established.

A note on the geometrical optics of diffraction by an interface

1964 ◽  
Vol 60 (4) ◽  
pp. 1013-1022 ◽  
Author(s):  
R. H. J. Grimshaw
Keyword(s):  
Wave Equation ◽  
Geometrical Optics ◽  
Cauchy Type ◽  
Order Partial Differential Equation ◽  
Normal Distance ◽  
The Cauchy Problem ◽  
Type Boundary ◽  
Image Position ◽  
Reflecting Surface ◽  
Homogeneous Media

1. It is well known that solutions of the Cauchy problem for the wave equation represent disturbances obeying the laws of geometrical optics. Specifically a solution ψ of the wave equationfor which ψ = δψ/δt = 0 initially outside a surface C0, vanishes at time t in the exterior of a surface Ct parallel to and at a normal distance ct from C0 (see e.g. (l), page 643). Analogous results hold for the solutions of any linear hyperbolic second-order partial differential equation with boundary-value conditions of the Cauchy type. Boundary conditions of the type representing reflexion have been treated by Friedlander(2). He showed that as well as the incident and reflected wavefronts, there sometimes exists a ‘shadow’ where diffraction occurs, and that the diffracted wave fronts are normal to the reflecting surface, the corresponding rays travelling along the surface and leaving it tangentially. The purpose of this paper is to extend these results to refraction, where instead of a purely reflecting surface we have an interface between two different homogeneous media.

scholarly journals Large diffusivity finite-dimensional asymptotic behaviour of a semilinear wave equation

2003 ◽  
Vol 2003 (8) ◽  
pp. 409-427 ◽  
Author(s):  
Robert Willie
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Wave Equation ◽  
Asymptotic Behaviour ◽  
Nonlinear Term ◽  
Damped Wave Equation ◽  
Order Ordinary Differential Equation ◽  
Finite Dimensional ◽  
Type Boundary ◽  
Large Diffusion

We study the effects of large diffusivity in all parts of the domain in a linearly damped wave equation subject to standard zero Robin-type boundary conditions. In the linear case, we show in a given sense that the asymptotic behaviour of solutions verifies a second-order ordinary differential equation. In the semilinear case, under suitable dissipative assumptions on the nonlinear term, we prove the existence of a global attractor for fixed diffusion and that the limiting attractor for large diffusion is finite dimensional.

scholarly journals A general blow-up result for a degenerate hyperbolic inequality in an exterior domain

2021 ◽  
Author(s):  
Mohamed Jleli ◽  
Mokhtar Kirane ◽  
Bessem Samet
Keyword(s):  
Boundary Conditions ◽  
Critical Exponent ◽  
Exterior Domain ◽  
Critical Case ◽  
Blow Up ◽  
Unified Approach ◽  
Dirichlet Type ◽  
Open Questions ◽  
Neumann Type ◽  
Type Boundary

In this paper, we consider a degenerate hyperbolic inequality in an exterior domain under three types of boundary conditions: Dirichlet-type, Neumann-type, and Robin-type boundary conditions. Using a unified approach, we show that all the considered problems have the same Fujita critical exponent. Moreover, we answer some open questions from the literature regarding the critical case.

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