Arched beams of Bresse type: observability and application in thermoelasticity

This is the first paper of a trilogy intended by the authors in what concerns a unified approach to the stability of thermoelastic arched beams of Bresse type under Fourier's law. Our main goal in this starting work is to develop an original observability inequality for conservative Bresse systems with non-constant coefficients. Then, as a powerful application, we prove mathematically that the stability of a partially damped model in thermoelastic Bresse beams is invariant under the boundary conditions. The exponential and optimal polynomial decay rates are addressed. This approach gives a new view on the stability of Bresse systems subject to different boundary conditions as well as it provides an accurate answer for the related issue raised by Liu and Rao (Z. Angew. Math. Phys. 60(1): 54-69, 2009) from both the physical and mathematical points of view.