Spectral Controllability of Some Singular Hyperbolic Equations On Networks

The purpose of this paper is to address the question of well-posedness and spectral controllability of the wave equation perturbed by potential on networks whcih may contain unbounded potentials in the external edges. It has been shown before that in the absence of any potential, there exists an optimal time T ∗ (which turns out to be simply twice the sum of all length of the strings of the network) that describes the spectral controllability of the system. We will show that this holds in our case too, i.e. the potentials have no effect on the value of the optimal time T ∗. The proof is based on the famous Beurling-Malliavin’s Theorem on the completeness interval of real exponentials and on a result by Redheffer who had shown that under some simple condition the completeness interval of two complex sequences are the same.