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

Yao, Pengfei

SIAM J. Control Optim. 37, No. 5, 1568-1599 (1999).

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.

Reviewer: S.Migorski (Krakow)

MSC:

35L20	Initial-boundary value problems for second-order hyperbolic equations
93B05	Controllability
78A05	Geometric optics

Keywords:

boundary exact controllability; Riemannian geometry method; geometric optics; multiplier identities