Observation problems posed for the Klein-Gordon equation

Transversal vibrations u = u(x,t) of a string of length l with fixed ends are considered, where u is governed by the Klein-Gordon equation u(tt)(x,t) = a(2)u(xx)(x,t), (x,t) is an element of [0, l] x R, a > 0, c < 0. Sufficient conditions are obtained that guarantee the solvability of each of four observation problems with given state functions f, g at two distinct time instants -infinity < t(1) < t(2) <infinity. The essential conditions are the following: smoothness of f, g as elements of a corresponding subspace D(s+i)(0,l) (introduced in [2]) of a Sobolev space H(s+i)(0, l), where i = 1,2 depending on the type of the observation problem, and the representability of t(2) - t(1) as a rational multiple of 2l/a. The reconstruction of the unknown initial data (u(x,0), u(t)(x,0)) as the elements of D(s+1)(0,l) x D(s)(0,l) are given by means of the method of Fourier expansion.