EXACT OBSERVABILITY OF THE TIME-VARYING HYPERBOLIC EQUATION WITH FINITELY MANY MOVING INTERNAL OBSERVATIONS

摘要

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(infinity)(O,T; R(n+1))- or C([O,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 [O, T] for n = 1) and set-valued maps (continuous on [O,T] with respect to Lebesgue measure) is established. The techniques involved are related to the construction of suitable skeletons for these curves and maps. [References: 22]