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Exact controllability of the wave equation with
mixed boundary condition and time-dependent coefficients

##
*
M. M. Cavalcanti
*

** Address.**
Departamento de Matematica - Universidade Estadual de
Maringa, 87020-900 Maringa - PR, BRAZIL.

** E-mail:**

**Abstract.**
In this paper we study the boundary exact
controllability for the equation
$$
\frac{\partial}{\partial t}\left(\alpha (t){{\partial y}\over {
\partial t}}\right)-\sum_{j=1}^n{{\partial}\over {\partial x_j}}\left
(\beta (t)a(x){{\partial y}\over {\partial x_j}}\right)=0\;\;\;\hbox{in}\;\;
\Omega\times (0,T)\,,
$$
when the control action is of Dirichlet-Neumann form and $\Omega$ is
a bounded domain in ${\bold R}^n$. The result is obtained
by applying the HUM (Hilbert Uniqueness Method) due to J. L. Lions.

**AMSclassification.**
35B40, 35B35, 35L99.

**Keywords.**
Wave equation, boundary value problem,
exact controllability, Dirichlet-Neumann condition.