U. Biccari, M. Warma Null-controllability properties of a fractional wave equation with a
memory term. Evol. Eq. Control The., Vol. 9, No. 2 (2020), pp. 399-430
Abstract: We study the null-controllability properties of a one-dimensional wave equation with memory associated with the fractional Laplace operator. The goal is not only to drive the displacement and the velocity to rest at some time-instant but also to require the memory term to vanish at the same time, ensuring that the whole process reaches the equilibrium. The problem being equivalent to a coupled nonlocal PDE-ODE system, in which the ODE component has zero velocity of propagation, we are required to use a moving control strategy. Assuming that the control is acting on an open subset ω(t) which is moving with a constant velocity c∈R , the main result of the paper states that the equation is null controllable in a sufficiently large time T and for initial data belonging to suitable fractional order Sobolev spaces. The proof will use a careful analysis of the spectrum of the operator associated with the system and an application of a classical moment method.