Existence, Uniqueness and Stabilization of Solutions of a Generalized Telegraph Equation on Star Shaped Networks
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Existence, Uniqueness and Stabilization of Solutions of a Generalized Telegraph Equation on Star Shaped Networks Alaa Hayek1,2 · Serge Nicaise2 Ali Wehbe1
· Zaynab Salloum1 ·
Received: 25 March 2020 / Accepted: 19 September 2020 © Springer Nature B.V. 2020
Abstract The existence, uniqueness, strong and exponential stability of a generalized telegraph equation set on one dimensional star shaped networks are established. It is assumed that a dissipative boundary condition is applied at all the external vertices and an improved Kirchhoff law at the common internal vertex is considered. First, using a general criteria of Arendt-Batty (see Arendt and Batty in Trans. Am. Math. Soc. 306(2):837–852, 1988), combined with a new uniqueness result, we prove that our system is strongly stable. Next, using a frequency domain approach, combined with a multiplier technique and the construction of a new multiplier satisfying some ordinary differential inequalities, we show that the energy of the system decays exponentially to zero. Keywords Telegraph equation · Strong stability · Exponential stability · Frequency domain approach
1 Introduction Over decades, telegraph equations have gained attention and interest among scientists due to their different applications in the transmission of electrical signals along transmission lines of all frequencies, in addition to many other physical, biological and engineering applications (see [4, 8, 17, 25, 31]). As a consequence, many mathematical models were set up, for
B S. Nicaise
[email protected] A. Hayek [email protected] Z. Salloum [email protected] A. Wehbe [email protected]
1
Faculty of sciences 1, Khawarizmi Laboratory of Mathematics and Applications-KALMA, Lebanese University, Hadath-Beirut, Lebanon
2
Université polytechnique Hauts-de-France (UPHF-LAMAV), Valenciennes Cedex 9, France
A. Hayek et al.
instance in [15], a general and realistic situation was considered and a mathematical model of electromagnetic wave propagation in heterogeneous lossy coaxial cables was derived. Recently, referring to [29], S. Nicaise has considered the stabilization of the generalized telegraph equation set in a real interval (model on a cable from [15]): ⎧ ⎪ ⎨Vt + gV + aIx + kW = 0, in (0, L) × (0, ∞), (1.1) It + rI + bVx = 0, in (0, L) × (0, ∞), ⎪ ⎩ in (0, L) × (0, ∞), Wt + cW = V , with the following boundary conditions V (0, t) = V (L, t) = 0, t ∈ R∗+ , and the following initial conditions V (x, 0) = V0 (x), I (x, 0) = I0 (x), W (x, 0) = W0 (x), x ∈ (0, L), where, a, b, c, r, k and g are all non-negative functions in L∞ (0, L) that verify some assumptions mentioned in [29], see (1.2) below for the exact conditions. The generalized telegraph equation is a coupling between the usual telegraph equation where the electric unknowns are V and I representing the electric potential and the electric current respectively with a first order differential equation of parabolic type involving an auxiliary variable W representing the non-local effects. In [29], the author was interested in s
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