Controllability, returning waves and scattering without reverberation in 3D acoustic dynamic system

Abstract

The dynamic acoustic scattering system is governed by the wave equation $u_{tt}-Δu+qu=0$ in $\Bbb R^3,\,\,\,-\infty<t<\infty$, with a compactly supported potential $q$ and infinitely distant sources (controls) $f$, which initiate incoming spherical waves $u=u^f(x,t)$ provided $u^f\big|_{|x|<-t,\,\,\,t<0}=0$. These waves are focused at $x=0$ and fill up the whole space at the moment $t=0$. The system is {\it controllable} if the set of waves $u^f(\cdot,0)$ produced by all finite energy controls $f$, covers the space $L_2(\Bbb R^3)$. As we show, if the Hamiltonian $H=-Δ+q$ has the bound states, then in the space the points $a$ appear such that the system, being refocused at $x=a$, loses controllability. The latter leads to a physical effect: the waves $u^f$ of finite energy appear, which vanish simultaneously in the past and future cones $|x|<\pm\, t$ and leave the region of inhomogeneity of $q$ without reverberation. This effect has some similarities with the wavefront reversal (Time Reversing Mirror), but is more meaningful from a mathematical point of view.