On the exponential and polynomial stability for a linear Bresse system

M Afilal (1) , A. Guesmia (2) , A Soufiyane (3) , M Zahri (3)

1 UCA - Université Cadi Ayyad [Marrakech]
2 IECL - Institut Élie Cartan de Lorraine
3 UoS - University of Sharjah

In this paper, we consider a linear one-dimensional Bresse system consisting of three hyperbolic equations coupled in a certain manner under mixed homogeneous Dirichlet-Neumann boundary conditions. Here, we consider that only the longitudinal displacement is damped, and the vertical displacement and shear angle displacement are free. We prove the well-posedness of the system and some exponential, lack of exponential and polynomial stability results depending on the coefficients of the equations and the smoothness of initial data. At the end, we use some numerical approximations based on finite difference techniques to validate the theoretical results. The proof is based on the semigroup theory and a combination of the energy method and the frequency domain approach.

Mots clés: symptotic behavior; Bresse system; energy method; frequency domain approach; well-posedness; semigroup theory; numerical approximation

Domaines: Equations aux dérivées partielles [math.AP]

M Afilal, A. Guesmia, A Soufiyane, M Zahri. On the exponential and polynomial stability for a linear Bresse system. Mathematical Methods in the Applied Sciences, 2020, 43 (5), pp.2626-2645. ⟨10.1002/mma.6070⟩. ⟨hal-02891562⟩