MiS Preprint Repository

Consider the problem of a compact, n-dimensional Riemannian manifold-with-boundary $\bar{\mathrm{\Omega }}$ and the natural hyperbolic P.D.E. (Riemannian wave equation): $$\frac{{\partial }^{2}u}{\partial t^2}={\mathrm{\Delta }}_{g}u,(1)$$ plus possible lower-order terms, where ${\mathrm{\Delta }}_g$ is the Riemannian Laplace operator, or Laplace-Beltrami operator, of $\mathrm{\Omega }$ We consider the problem of the control in time T of the wave equation from the boundary $\partial \mathrm{\Omega }$ of $\mathrm{\Omega }$ by specifying Dirichlet boundary controls on $\partial \mathrm{\Omega }\times [0,T]$ The question we address is whether, for any Cauchy data on $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$ we pose the question: are chords unique? Here, a chord is a length-minimizing geodesic of $\bar{\mathrm{\Omega }}$ joining two given points of $\partial \mathrm{\Omega }$ We assume that any two points of $\partial \mathrm{\Omega }$ are connected by at most one (and hence exactly one) chord.
If, in addition, the chords are nondegenerate and $\partial \mathrm{\Omega }$ has positive second fundamental form, then the wave equation is controllable from $\partial \mathrm{\Omega }$ in any time T greater than the maximum distance in $\bar{\mathrm{\Omega }}$ between points of $\partial \mathrm{\Omega }$
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 $\mathrm{\Omega }$

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.