Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems

Gervy Marie Angeles; Gilbert Peralta

doi:10.3934/eect.2020108

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2022, Volume 11, Issue 1: 199-224. Doi: 10.3934/eect.2020108

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Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems

Gervy Marie Angeles^1,2, and
Gilbert Peralta^2, ,

1.
Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
2.
Department of Mathematics and Computer Science, University of the Philippines Baguio, Governor Pack Road, Baguio, 2600 Philippines

^* Corresponding author
^* Corresponding author

Received: May 31, 2020

Revised: September 30, 2020

Early access: December 2020

Published: February 2022

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Abstract

We consider a hyperbolic system of partial differential equations on a bounded interval coupled with ordinary differential equations on both ends. The evolution is governed by linear balance laws, which we treat with semigroup and time-space methods. Our goal is to establish the exponential stability in the natural state space by utilizing the stability with respect to the first-order energy of the system. Derivation of a priori estimates plays a crucial role in obtaining energy and dissipation functionals. The theory is then applied to specific physical models.

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Mathematics Subject Classification: Primary: 35L50, 47D03; Secondary: 93D20.

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Figure 1. An elastic tube connected to two rigid tanks

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Figure 2. A waveguide terminated by oscillators

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