Pointwise and internal controllability for the wave equation (Q1401581)

The authors consider the observation and control of the one-dimensional wave equation on \(\varphi = [0,\ell]\times [0,T]\) with observation and control restricted to a small spatial subinterval \([0,0+\varepsilon]\subset (0,\ell)\). The main interest is in the asymptotics as \(\varepsilon\to 0,\) i.e., as the subinterval shrinks to the single point \(\{a\}\). In previous papers, quoted by the authors, results were obtained showing an essential dependence on number-theoretic properties of \(a/\ell\) -- with a distinction between ``good'' points for which the asymptotics behave well and ``bad'' points for which they do not. The principal novelty of the paper is the introduction of a natural contextual change which makes the asymptotics behave well for all choices of the point \(a\in (0,\ell)\).