Sharp uniform stabilization of a shallow shell with internal rotational inertia dissipation and a nonlinear boundary feedback under free boundary conditions

The motion of a shell is characterized by a system of two coupled hyperbolic partial differential equations: (i) an elastic wave in the two-dimensional in-plane displacement, and (ii) a plate-like Kirchhoff equation in the scalar normal displacement. A dynamic shallow shell is defined on a two-dimensional bounded, smooth manifold with elastic deformations accounting for vertical and in-plane oscillations. In this paper, we investigate the uniform stabilization of a dynamic shallow shell model, which is subject to nonlinear dissipation on part of its boundary under absorbing boundary conditions corresponding to the wave equation, while the plate equation incorporates rotary inertia terms along with inertial dissipation and inertial boundary spill-over. The model is motivated by applications to flutter suppression [1] in bridges and buildings, where the flutter occurs due to the interaction between the flow of gas and the structure. The uniform stabilization for a shell model is established by energy method in the Riemannian geometric setting and microlocal analysis techniques. The geometric approach offers an intrinsic, coordinate - independent framework along with an initial observability-type inequality. Meanwhile, microlocal analysis provides the needed sharp trace estimates crucial for solving the stabilization problem concerning the elastic wave. Moreover, it enables the removal of geometric constraints on the controlled portion of the boundary. The final result provides uniform decay rates of the energy which are quantitative via an ODE algorithm based on the behavior of the boundary dissipation at the origin. Mathematics Subject Classification (2020). 35B35; 74K25; 93C20; 93D15; 93D20.