Internal exact controllability and uniform decay rates for a model of dynamical elasticity equations for incompressible materials with a pressure term

Abstract

This paper is concerned with the internal exact controllability of the following model of dynamical elasticity equations for incompressible materials with a pressure term, $$\phi''-\Delta \phi=-\nabla p,$$ and it is also devoted to the study of the uniform decay rates of the energy associated with the same model subject to a locally distributed nonlinear damping, $$\phi''-\Delta \phi+a(x)g(\phi')=-\nabla p,$$ where $\Omega$ is a bounded connected open set of $\mathbb{R}^n$ $(n\geq 2)$ with regular boundary $\Gamma,$ $\phi=(\phi_1(x,t),\dots, \phi_n(x,t)),$ $x=(x_1,\dots,x_n)$ are $n$-dimensional vectors and $p$ denotes a pressure term. The function $a(x)$ is assumed to be nonnegative and essentially bounded and, in addition, $a(x)\geq a_0>0$ a.e. in $\omega\subset \Omega$, where $\omega$ satisfies the geometric control condition. The first result is obtained by applying HUM (Hilbert Uniqueness Method) due to J. L. Lions while the second one is obtained by employing ideas first introduced in the literature by Lasiecka and Tataru.

Keywords

• internal exact controllability
• incompressible materials
• non linear damping
• uniform decay rates