On boundary exact controllability of one‐dimensional wave equations with weak and strong interior degeneration

Abstract

In this paper, we study exact boundary controllability for a linear wave equation with strong and weak interior degeneration of the coefficient in the principle part of the elliptic operator. The objective is to provide a well‐posedness analysis of the corresponding system and derive conditions for its controllability through boundary actions. Passing to a relaxed version of the original problem, we discuss existence and uniqueness of solutions, and using the HUM method we derive conditions on the rate of degeneracy for both exact boundary controllability and the lack thereof.

Boundary controllability for a degenerate and singular wave equation

“… As a first thing, we point out that combining our proofs with the ideas of Vancostenoble, 9 a possible way to improve our results would be to study the case of a degenerate/singular operator with $$$ \frac{\mu }{x^{\beta }} $$$ with $$$ \beta \le 2-\alpha $$$ instead of $$$ \frac{\mu }{x^{2-\alpha }} $$$. Alabau‐Boussouira et al 10 treat the case of a degenerate operator $$$ {\left(a(x){u}_x\right)}_x $$$ with a general coefficient $$$ a(x) $$$ that vanishes at $$$ x=0 $$$. It would be interesting to consider a simultaneously degenerate and singular equation with a general degenerate inhomogeneous speed and general singular potential. Inspired by the results in Kogut et al, 14 it would also be interesting to study the wave equation with degeneracy and singularity at the interior of the space domain as done in Fragnelli and Mugnai 32 for the heat equation. Finally, the study of null controllability properties of degenerate or singular coupled wave equations is still to be done and many further directions remain to be investigated. …”

“…In this case, controllability properties by means of a boundary control have been investigated in various papers. We refer the reader to the following pioneering contributions 10–15 . We also refer to Zhang and Gao 16,17 for other works on controllability problems by means of a locally distributed control.…”

Indirect Boundary Controllability of Coupled Degenerate Wave Equations

“…Recently, several controllability results have also been obtained for scalar degenerate equations, see for example [24,25,26,27,28,29,30,31,32,33]. However, in the more complex situation of coupled degenerate hyperbolic equations, not many things are known.…”

On Boundary Null Controllability of Strongly Degenerate Hyperbolic Systems on Star-Shaped Planar Network

“…While parabolic controllability problems with degeneration have been discussed in detail in e.g. [2], wave equations with degeneration in the leading coefficients are much less explored, see however [3,10] for problems in which damage occurs at the boundary of the domain, and [15] where the 'damaged' point is internal. Very recently, in-span damage has been considered for the 1-d parabolic equation in [4].…”