Appl Math Optim 46:167–178 (2002)
2002 Springer-Verlag New York Inc.
Boundary Control of PDEs via Curvature Flows:
the View from the Boundary, II
Santiago Betel´u, Robert Gulliver, and Walter Littman
School of Mathematics
Institute for Mathematics and its Applications,
University of Minnesota,
Minneapolis, MN 55455, USA
Abstract. We describe some results on the exact boundary controllability of
the wave equation on an orientable two-dimensional Riemannian manifold with
nonempty boundary. If the boundary has positive geodesic curvature, we show that
the problem is controllable in ﬁnite time if (and only if) there are no closed geodesics
in the interior of the manifold. This is done by solving a parabolic problem to con-
struct a convex function. We exhibit an example for which control from a subset of
the boundary is possible, but cannot be proved by means of convex functions. We
also describe a numerical implementation of this method.
Key Words. Boundary control, Wave equation, Riemannian manifold, Curvature
ﬂow, Pseudoconvex function.
AMS Classiﬁcation. 35Lxx, 53C44, 93C20, 35B37.
On a Riemannian manifold (
, g), for a linear hyperbolic partial differential equation
u + lower-order terms, (1)
× [0, T ] → R, the problem of boundary controllability is of widespread
interest. By this, we mean the question whether, for any given initial data (u