×

On the observability inequalities for exact controllability of wave equations with variable coefficients. (English) Zbl 0951.35069

The paper deals with the problem of boundary exact controllability for the mixed problem for the wave equation \[ \begin{cases} y_{tt} + Ay = 0 & \text{ in } (0,T),\\ y(x,0) = y^0(x), \\ y_t(x,0) = y^1(x) & \text{ in } \Omega,\\ y = 0 & \text{ in } \partial_1 \Omega \times (0,T),\\ y = v & \text{ in } \partial_2 \Omega \times (0,T)\text{ or } {{\partial y}\over {\partial {\nu_A}}} = v \text{ in } \partial_2 \Omega \times (0,T),\end{cases} \] where the operator \( Ay = \sum_{i,j=1}^n D_i (a_{ij}(x) D_j y)\) has uniformly elliptic \(C^\infty\) coefficients. It is well know that this problem is equivalent to the observability inequality. The latter is proved in this paper by the Riemannian geometry method under some geometric conditions by using several multiplier identities.

MSC:

35L20 Initial-boundary value problems for second-order hyperbolic equations
93B05 Controllability
78A05 Geometric optics
PDF BibTeX XML Cite
Full Text: DOI