Chord uninqueness and controllability: the view from the boundary, I
Robert Gulliver and Walter Littman
Contact the author: Please use for correspondence this email.
Submission date: 21. Sep. 2000
published in: Differential geometric methods in the control of partial differential equations / R.Gulliver (ed.)
Providence, RI : American Math. Soc., 2000. - P. 145 - 176
(Contemporary mathematics ; 268)
Download full preprint: PDF (520 kB), PS ziped (216 kB)
Consider the problem of a compact, n-dimensional Riemannian manifold-with-boundary and the natural hyperbolic P.D.E. (Riemannian wave equation):
plus possible lower-order terms, where is the Riemannian Laplace operator, or Laplace-Beltrami operator, of We consider the problem of the control in time T of the wave equation from the boundary of by specifying Dirichlet boundary controls on The question we address is whether, for any Cauchy data on 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 we pose the question: are chords unique? Here, a chord is a length-minimizing geodesic of joining two given points of We assume that any two points of are connected by at most one (and hence exactly one) chord.
If, in addition, the chords are nondegenerate and has positive second fundamental form, then the wave equation is controllable from in any time T greater than the maximum distance in between points of
This result provides a counterpoint to controllability theorems such as those in ,  and , 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
- 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. Applications235 (1999), 13-57.
Tataru, Daniel: Boundary controllability of conservative PDEs. Appl. Math. Optim.31 (1995), 257-295.