Energy decay for the damped wave equation on an unbounded network

Abstract

We study the wave equation on an unbounded network of N, N ∈ N * , finite strings and a semi-infinite one with a single vertex identified to 0. We consider continuity and dissipation conditions at the vertex and Dirichlet conditions at the extremities of the finite edges. The dissipation is given by a damping constant α > 0 via the condition N j=0 ∂xu j (0, t) = α∂tu 0 (0, t). We give a complete spectral description and we use it to study the energy decay of the solution. We prove that for α = N + 1 we have an exponential decay of the energy and we give an explicit formula for the decay rate when the finite edges have the same length.

Stability of an infinite star‐shaped network of strings by a Kelvin–Voigt damping

“…One of the most interesting question is about the energy of a wave that propagates along a network of strings bounded or unbounded. 11 Hereafter, we give the spectral properties of the operator . We describe the spectrum and investigate the strong stability of the C 0 semigroup associated to the system (1), using a criteria of Arendt-Batty.…”

Stability of Linear KdV Equation in a Network with Bounded and Unbounded Lengths

“…We also mention [9] where controllability properties were studied and [10], [11] where the exponential stability was achieved by acting with damping terms with time-delay and saturation, respectively (see [12] for more problems related to KdV in networks). The main difference of this work with the previously cited is that, we consider a star-shaped network mixing bounded and unbounded lengths as for example [13], [14] in the case of wave equation. With respect to the KdV equation defined on the halfline, we can mention, for instance, [5], [15] which focus on the well-posedness properties.…”

Bilinear quantum systems on compact graphs: Well-posedness and global exact controllability

“…Different results were developed by addressing the problem directly or by duality (see for instance [Bur91,Leb92,Lio83,LT92]). For results on networks, we refer to [AG18,AJ04,AJK05,MAN17] and to [AN15,DZ06]. Regarding inverse problems, we cite [ALM10,Bel04] for the boundary control approach and [BCV11,IPR12] for uniqueness and stability results via Carleman estimates.…”