Exact Observability of the Time-Varying Hyperbolic Equation with Finitely Many Moving Internal Observations

The problem of exact observability of the linear hyperbolic equation with time-varying coefficients under finitely many internal observations is considered. The question with which we are concerned in this paper is a sharp correspondence between the internal regularity of the solutions and a type of observation required to provide $L^\infty(0,T; R^{n+1})$- or $C([0,T[; R^{n+1})$-exact observability with respect to the energy norm. Two types of observations are considered: pointwise and spatially averaged, for which the existence of needed observation curves (continuous on $[0,T[$ for $n=1$) and set-valued maps (continuous on $[0,T[$ with respect to Lebesgue measure) is established. The techniques involved are related to the construction of suitable skeletons for these curves and maps.