Existence, Uniqueness and Stabilization of Solutions of a Generalized Telegraph Equation on Star Shaped Networks

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.

Stability and asymptotic properties of dissipative evolution equations coupled with ordinary differential equations

“…The generalized telegraph equation on networks. In the present subsection, motivated by [28,29,44,45,26], we consider the generalized telegraph equation on a C 2 -network N . Let us then shortly recall the notion of C 2 -networks, which is simply those of [53] (we also refer to [2,3,52,54,8,38,41] for more details).…”

“…where A is an unbounded operator that is the generator of a C 0 semigroup in the Hilbert space H, B is a bounded operator from another Hilbert space X, and M , N are supposed to be bounded operators. Many problems from physics enter in this framework, let us mention dispersive medium models [32,34,55,43,46,31], the generalized telegraph equations [28,29,44,26], the heat equation with memory effects [20,50,35], and cascades of ODE-hyperbolic systems [30,36], see below for more explanations. Eliminating P by the formula P (t) = e Bt P 0 + t 0 e B(t−s) U (s) ds, we see that the first equation of (1.1) becomes U t = Au + M e Bt P 0 + t 0 M e B(t−s) U (s) ds.…”