Exact controllability of a second order integro-differential equation with a pressure term

Abstract

This paper is concerned with the boundary exact controllability of the equation $$u''-\Delta u-\int_0^t g(t-\sigma)\Delta u(\sigma) d\sigma=-\nabla p$$ where $Q$ is a finite cilinder $\Omega\times]0,T[$, $\Omega$ is a bounded domain of $R^n$, $u=(u_1(x,t),\ldots,u_n(x,t))$, $x=(x_1,\ldots,x_n)$ are $n$-dimensional vectors and $p$ denotes the pressure term. The result is obtained by applying HUM (Hilbert Uniqueness Method) due to J. L. Lions. The above equation is a simple model of dynamical elasticity equations for incompressible materials with memory.