Consider a hyperbolic PDE with smooth, time-independent coefficients in M x [0,T], where M is a smooth, relatively compact domain in real n-space or is a smooth n-dimensional manifold with boundary bd M: u_tt = a_iju_ij + lower-order terms. We use the coefficients a_ij to define a Riemannian metric ds² = g_ij dxⁱ dx^j, where for each x in M, (g_ij(x)) is the inverse of the matrix (a_ij(x)). Modulo first-order terms, the right-hand side of the PDE is the Riemannian Laplace operator. We use concepts such as convexity and geodesic curvature in their Riemannian, and hence coordinate-independent, versions.

The boundary control problem is whether, for any initial conditions at t = 0, there is a choice of (say) Dirichlet boundary values on (bd M) x [0,T] so that the solution of the PDE vanishes for t >= T. Write T₀ for the minimum value of T for which this is possible. It has been shown (under strong smoothness hypotheses: e.g. Bardos--Lebeau--Rauch) that T₀ is the maximum length of geodesics in M.

We will outline a few recent results: (1) If there is a convex function v:M --> [0,K] with Hessian matrix greater than 2c ds², then T₀ <= 2 sqrt(K/c); (2) If bd M is locally convex, and if the minimizing geodesic joining any two points of bd M is unique and nondegenerate, then T₀ is the maximum of the distance between boundary points; and (3) If n = 2, bd M is locally convex and there are no closed geodesics in M, then T₀ is finite, with an estimate. We will emphasise result (3), which uses Grayson's work on the flow of curves by curvature.

(1) is due to Lasiecka--Triggiani--Yao, with a new proof by Michael Galbraith; (2) is joint work with Walter Littman; and (3) is joint work with Littman and Santiago Betel\'u.