Chord uniqueness and controllability: The view from the boundary. I (Q2708678)

The authors consider a problem important for compact, \(n\)-dimensional Riemannian manifolds with boundary and the natural hyperbolic P.D.E: (Riemannian wave equation). They study the controllability in the border of a domain \(D\). To reduce the question of the controllability to the computable problem about geodesics on \(D\), we pose the question: are chords unique? Here, a chord is a length-minimizing geodesic on to chord of \(D\) joining two given points of the boundary of \(D\). We assume that any two points of boundary of \(D\) are connected by at most one (and hence excactly one) chord. Finally those results provide the counterpoint to controllability theorems as those in [\textit{D. Tataru}, Appl. Math. Optimization 31, No. 3, 257-295 (1995; Zbl 0836.35085); \textit{I. Lasiecka, R. Triggiani} and \textit{P.-F. Yao}, Nonlinear Anal. Theory Methods Appl. 30, 111-122 (1997; Zbl 0904.35045), and J. Math. Anal. Appl. 235, No. 1, 13-57 (1999; Zbl 0931.35022)], in which the existence of the convex function, and hence roughly speaking an upper bound on sectional curvature, is assumed. We require the direct hypothesis on the Riemannian metric in the interior of \(D\). NEWLINENEWLINENEWLINEThe paper is a great contribution for the study of the wave equation in a compact, \(n\)-dimensional Riemannian manifold. Besides, the techniques used by the authors can be used to obtain results for other P.D.E.NEWLINENEWLINEFor the entire collection see [Zbl 0956.00042].