MiS Preprint Repository

Consider the problem of a compact, n-dimensional Riemannian manifold-with-boundary $\overline{\Omega}$ and the natural hyperbolic P.D.E. (Riemannian wave equation): $$\frac{\partial^2 u}{\partial t^2}=\Delta_g u, \ \ \ (1)$$ plus possible lower-order terms, where $\Delta_g$ is the Riemannian Laplace operator, or Laplace-Beltrami operator, of $\Omega$ We consider the problem of the control in time T of the wave equation from the boundary $\partial \Omega$ of $\Omega$ by specifying Dirichlet boundary controls on $\partial \Omega \times [0,T]$ The question we address is whether, for any Cauchy data on $\Omega$ at the initial time t = 0, there is a choice of boundary control which will achieve any prescribed Cauchy data at the terminal time t=T.
In order to reduce this controllability question to a computable problem about geodesics on $\Omega$ we pose the question: are chords unique? Here, a chord is a length-minimizing geodesic of $\overline{\Omega}$ joining two given points of $\partial \Omega$ We assume that any two points of $\partial \Omega$ are connected by at most one (and hence exactly one) chord.
If, in addition, the chords are nondegenerate and $\partial \Omega$ has positive second fundamental form, then the wave equation is controllable from $\partial \Omega$ in any time T greater than the maximum distance in $\overline{\Omega}$ between points of $\partial \Omega$
This result provides a counterpoint to controllability theorems such as those in [3], [1] and [2], in which the existence of a convex function, and hence - roughly speaking - an upper bound on sectional curvature, is assumed. We require no direct hypothesis on the Riemannian metric in the interior of $\Omega$

Lasiecka, Irene, Roberto Triggiani and Peng-Fei Yao: Exact controllability for second-order hyperbolic equations with variable coefficient-principal part and first-order terms. Nonlinear Anal. 30 (1997), 111-122.

Lasiecka, Irene, Roberto Triggiani and Peng-Fei Yao: Inverse/observability estimates for second-order hyperbolic equations with variable coefficients. J. Math. Anal. Applications 235 (1999), 13-57.

Tataru, Daniel: Boundary controllability of conservative PDEs. Appl. Math. Optim. 31 (1995), 257-295.