Preprint 58/2000

Chord uninqueness and controllability: the view from the boundary, I

Robert Gulliver and Walter Littman

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Submission date: 21. Sep. 2000
Pages: 33
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) 
Bibtex
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Abstract:
Consider the problem of a compact, n-dimensional Riemannian manifold-with-boundary img43 and the natural hyperbolic P.D.E. (Riemannian wave equation):
equation4

plus possible lower-order terms, where img45 is the Riemannian Laplace operator, or Laplace-Beltrami operator, of img47 We consider the problem of the control in time T of the wave equation from the boundary img51 of img53 by specifying Dirichlet boundary controls on img55 The question we address is whether, for any Cauchy data on img57 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 img53 we pose the question: are chords unique? Here, a chord is a length-minimizing geodesic of img43 joining two given points of img67 We assume that any two points of img51 are connected by at most one (and hence exactly one) chord.

If, in addition, the chords are nondegenerate and img51 has positive second fundamental form, then the wave equation is controllable from img51 in any time T greater than the maximum distance in img43 between points of img67

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 img47

  1. 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.

  2. 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.

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

 

23.06.2018, 02:10