The Exact Bounded Solution to an Initial Boundary Value Problem for 1D Hyperbolic Equation with Interior Degeneracy. I. Separation of Variables

“…Remark It is clear that the proposed relaxation is only a matter of regularity of some solutions. We refer to the recent papers, 23–25 where the authors consider a particular case of the problem ()–() with

a false(x false) = const false| x - 1 {false|}^{α}

for α ∈ [0, 2), and they show that this problem can admit many solutions, but only one of them satisfies transmission conditions ()–() and has a continuously differentiable flux at

x = 1

. As for the rest ones, they satisfy transmission conditions in the form ()–().…”

Section: Preliminariesmentioning

confidence: 99%

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

Kogut

Kupenko

Leugering

2021

Math Methods in App Sciences

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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.

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“…It is clear that the proposed relaxation is only a matter of regularity of some solutions. We refer to the recent papers [2,3], where the authors consider a particular case of the problem (1.4)-(1.6) with a(x) = const |x − 1| α for α ∈ [1, 2), and they show that this problems is ill-posed and admits many solutions, but only one of them satisfies transmission conditions (2.51)-(2.52) and has a continuously differentiable flux at x = 1. As for the rest ones, they satisfy transmission conditions in the form (2.53)-(2.54).…”

Section: ω) Our Next Intention Is To Show That Due To the Properties ...mentioning

confidence: 99%

On Boundary Exact Controllability of One-Dimensional Wave Equations with Weak and Strong Interior Degeneration

Kogut,

Kupenko,

Leugering

2021

Preprint

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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.

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The Exact Bounded Solution to an Initial Boundary Value Problem for 1D Hyperbolic Equation with Interior Degeneracy. I. Separation of Variables

Cited by 6 publications

References 8 publications

Can a Finite Degenerate ‘String’ Hear Itself? Numerical Solutions to a Simplified IBVP

Can a Finite Degenerate ‘String’ Hear Itself? Numerical Solutions to a Simplified IBVP

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

On Boundary Exact Controllability of One-Dimensional Wave Equations with Weak and Strong Interior Degeneration

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