Geodesic
Exact controllability of the wave equation with mixed boundary condition and time-dependent coefficients
Archivum mathematicum, Tome 35 (1999) no. 1, pp. 29-57.

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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 ${R}^n$. The result is obtained by applying the HUM (Hilbert Uniqueness Method) due to J. L. Lions.
Classification : 35B35, 35B40, 35L05, 35L99, 93B05, 93C20
Keywords: wave equation; boundary value problem; exact controllability; Dirichlet-Neumann condition 
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     title = {Exact controllability of the wave equation with mixed boundary condition and time-dependent coefficients},
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Cavalcanti, M. M. Exact controllability of the wave equation with mixed boundary condition and time-dependent coefficients. Archivum mathematicum, Tome 35 (1999) no. 1, pp. 29-57. http://geodesic.mathdoc.fr/item/ARM_1999__35_1_a3/