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Stability results of a distributed problem involving Bresse system with history and/or Cattaneo law under fully Dirichlet or mixed boundary conditions
Author(s) -
Abdallah Farah,
Ghader Mouhammad,
Wehbe Ali
Publication year - 2018
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4717
Subject(s) - dirichlet boundary condition , mathematics , exponential stability , mathematical analysis , boundary value problem , thermal conduction , boundary (topology) , stability (learning theory) , polynomial , displacement (psychology) , physics , nonlinear system , thermodynamics , computer science , quantum mechanics , machine learning , psychology , psychotherapist
In this paper, we study the stability of a 1‐dimensional Bresse system with infinite memory‐type control and/or with heat conduction given by Cattaneo's law acting in the shear angle displacement. When the thermal effect vanishes, the system becomes elastic with memory term acting on one equation. We consider the interesting case of fully Dirichlet boundary conditions. Indeed, under equal speed of propagation condition, we establish the exponential stability of the system. However, in the natural physical case when the speeds of propagation are different, using a spectrum method, we show that the Bresse system is not uniformly exponentially stable. In this case, we establish a polynomial energy decay rate. Our study is valid for all other mixed boundary conditions.