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.