Recently, I have gotten involved with **geometric methods for the boundary
control of hyperbolic PDE's.**

- The first paper in this direction
is joint work with Walter Littman
[40], Postscript version or
PDF version.
We discovered that the wave equation in a Riemannian manifold M with boundary
may be controlled exactly from the boundary in any time greater than the
maximum distance between boundary points, provided that the boundary curves
inward and any two boundary points are joined by a unique minimizing curve,
which is free of conjugate points. This time of control is optimal. Numerous
novel examples are given.

- Assume that the
boundary of M is nonempty and has inward curvature. In the
**two-dimensional**case, Santiago Betelu, Littman and I have established that the well-known*necessary*condition for exact controllability from the boundary, that there be no closed geodesics in M, is also*sufficient.*See the Postscript version or the PDF version of [43].

In**dimension n,**the condition that there be no closed geodesics needs to be strengthened: instead, one should assume that there are no compact Gauss-flat submanifolds of any dimension between 1 and n-1. This is work in progress ([58]) with Walter Littman, based on new results of Ben Andrews on harmonic mean-curvature flow.

- A survey of some of these results, and some illustrative examples,
are in a joint paper with Walter Littman; see the
Postscript version or
PDF version of [52], along with the
figure, which is not in these Postscript and PDF versions.

- In joint work with Irena Lasiecka, Walter Littman and Roberto Triggiani, we have outlined a number of geometric techniques for constructing convex functions on Riemannian manifolds (including a nice result of Burago and Zalgaller), and applied them to the control of a coupled hyperbolic system which describes a structural acoustic chamber, as well as to scalar wave equations. See the Postscript version or PDF version of [46].

[40].
Chord Uniqueness and Controllability: the View from the Boundary, I
(with Walter Littman),
* Contemporary Mathematics * **268**, 145-176 (2000).
Postscript version or
PDF version

[43]. Boundary Control of PDEs via Curvature Flows: the View
from the Boundary II (with Santiago Betelu and Walter Littman),
* Applied Math. & Optimization* ** 46,** 167-178
(2002).
Postscript version or the
PDF version.

[46]. The Case for Differential Geometry in the Control of
Single and Coupled PDEs: The Structural Acoustic Chamber
(with Irena Lasiecka, Walter Littman and Roberto Triggiani).
*IMA Volumes in Mathematics and its Applications*
** 137,** 73-182 (2003)
Postscript version or the
PDF version.

[52]. The Use of Geometric Tools in the Boundary Control of Partial
Differential Equations (with Walter Littman). Proceedings 3rd
ISAAC Conference (Berlin, 2001).
Postscript version or the
PDF version. 88 pages.
The
figure is missing from these versions.

[58]. Boundary Control of Wave Equations in high dimensions via
Harmonic Mean-Curvature Flow (with Walter Littman), in
preparation.