Control From an Interior Hypersurface

2017/11/10 by Galkowski, Jeffrey, Léautaud, Matthieu
#Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1711.03939

Abstract

We consider a compact Riemannian manifold M (possibly with boundary) and Σ⊂ M∖ ∂ M an interior hypersurface (possibly with boundary). We study observation and control from Σ for both the wave and heat equations. For the wave equation, we prove controllability from Σ in time T under the assumption (TGCC) that all generalized bicharacteristics intersect Σ transversally in the time interval (0,T). For the heat equation we prove unconditional controllability from Σ. As a result, we obtain uniform lower bounds for the Cauchy data of Laplace eigenfunctions on Σ under TGCC and unconditional exponential lower bounds on such Cauchy data.

Related

• Wave maps from circle to Riemannian manifold: global controllability is equivalent to homotopy
• Measure propagation along a \mathscrC⁰-vector field and wave controllability on a rough compact manifold
• Control for Schrödinger equation on hyperbolic surfaces
• Geometric control condition for the wave equation with a time-dependent observation domain
• Tunneling estimates and approximate controllability for hypoelliptic equations