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

M. M. Cavalcanti, Universidade Estadual de Maringá, Brasil
V. N. D. Cavalcanti, Universidade Estadual de Maringá, Brasil
A. Rocha, Universidade Federal do Rio de Janeiro, Brasil
J. A. Soriano, Universidade Estadual de Maringá, Brasil

E. J. Qualitative Theory of Diff. Equ., No. 9. (1998), pp. 1-18.

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.

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