The Use of Geometric Tools in the Boundary Control of Partial Differential Equations

There has been new interest in the successful application of differential geometric methods in the control of p.d.e.’s. (See for example Contemporary Mathematics #268, AMS 2000, particularly the article by the present authors). Here we describe those results, and some newer results using the methods of curvature flows. We also present an example for which control is possible but cannot be proved by means of any convex function. Although the subject of boundary control of partial differential equations is about a quarter of a century old, and that of Riemannian geometry much older still, until recently there has been relatively little interaction between the two. This is especially surprising in view of the rôle bicharacteristics play in boundary control, which naturally bring to mind geodesics— a basic concept in Riemannian geometry. We will describe some recently established links between the two subjects. Our focus in this paper is on the qualitative relationship between Riemannian geometry and boundary control. Thus we shall not attempt here to express controllability in terms of the optimal choice of Sobolev spaces, leaving such questions to other papers such as [7]; nor shall we attempt to find the optimal smoothness of the Riemannian metric and of other coefficients of the hyperbolic equation. Consider a compact, n-dimensional Riemannian manifold-with-boundary Ω.We assume that ∂Ω is smooth and nonempty, and that the metric of Ω is smooth, i.e., C∞. We are interested in the boundary control of the following natural hyperbolic partial differential equation (Riemannian wave equation) on Ω× [0, T ] :