One-dimensional wave equation with set-valued boundary damping: well-posedness, asymptotic stability, and decay rates

This paper is concerned with the analysis of a 1D wave equation $z_{tt}-z_{xx}=0$ on $[0,1]$ with a Dirichlet condition at $x=0$ and a damping acting at $x=1$ of the form $(z_t(t,1),-z_x(t,1))\in\Sigma$ for $t\geq 0$, where $\Sigma$ is a given subset of $\mathbb R^2$. The study is performed within an $L^p$ functional framework, $p\in [1, +\infty]$. We determine conditions on $\Sigma$ ensuring existence and uniqueness of solutions of that wave equation, its strong stability and uniform global asymptotic stability of the solutions. In the latter case, we study the corresponding decay rates and their optimality. We first establish a correspondence between the solutions of that wave equation and the iterated sequences of a discrete-time dynamical system in terms of which we investigate the above mentioned issues. This enables us to provide a necessary and sufficient condition on $\Sigma$ ensuring existence and uniqueness of solutions of the wave equation and an efficient strategy for determining optimal decay rates when $\Sigma$ verifies a generalized sector condition. In case the boundary damping is subject to perturbations, we derive sharp results regarding asymptotic perturbation rejection and input-to-state issues.

The Exact Frequency Equations for the Euler-Bernoulli Beam Subject to Boundary Damping

The Euler-Bernoulli (E-B) beam is the most commonly utilized model in the study of vibrating beams. The exact frequency equations for this problem, subject to energy-conserving boundary conditions, are well-known; however, the corresponding dissipative problem has been solved only approximately, via asymptotic methods. These methods, of course, are not accurate when looking at the low end of the spectrum. Here, we solve for the exact frequency equations for the E-B beam subject to boundary damping. Numerous numerical examples are provided, showing plots of both the complex wave numbers and the exponential damping rates for the first five frequencies in each case. Some of these results are surprising.

On a Laminated Timoshenko Beam with Nonlinear Structural Damping

In the present work, we study a one-dimensional laminated Timoshenko beam with a single nonlinear structural damping due to interfacial slip. We use the multiplier method and some properties of convex functions to establish an explicit and general decay result. Interestingly, the result is established without any additional internal or boundary damping term and without imposing any restrictive growth assumption on the nonlinear term, provided the wave speeds of the first equations of the system are equal.

The Exact Frequency Equations for the Rayleigh and Shear Beams with Boundary Damping

While the Euler-Bernoulli beam is the most commonly utilized model in studying vibrating beams, one often requires a model that captures the additional effects of rotary inertia or deformation due to shear. The Rayleigh beam improves upon the Euler-Bernoulli by including the former effect, while the shear beam is an improvement that includes the latter. While all of these problems have been well studied when subject to energy-conserving boundary conditions, none have been solved for the case of boundary damping. We compute the exact frequency equations for the Rayleigh and shear beams, subject to boundary damping and, in the process, we find interesting connections between the two models, despite their being very different.