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In this paper we introduce some new concepts for second-order hyperbolicequations: two-point boundary value problem, global exact controllability andexact controllability. For several kinds of important linear and nonlinear waveequations arising from physics and geometry, we prove the existence of smoothsolutions of the two-point boundary value problems and show the global exactcontrollability of these wave equations. In particular, we investigate thetwo-point boundary value problem for one-dimensional wave equation defined on aclosed curve and prove the existence of smooth solution which implies the exactcontrollability of this kind of wave equation. Furthermore, based on this, westudy the two-point boundary value problems for the wave equation defined on astrip with Dirichlet or Neumann boundary conditions and show that the equationstill possesses the exact controllability in these cases. Finally, as anapplication, we introduce the hyperbolic curvature flow and obtain a resultanalogous to the well-known theorem of Gage and Hamilton for the curvature flowof plane curves.

Two-point boundary value problems and exact controllability for several kinds of linear and nonlinear wave equations