The Controllability Problem for Abstract Wave Equations and Its Applications

The paper is devoted to the boundary controllability of the abstract wave equations when the control is exerted on a part of the boundary by means of one control. We give a Kalman-type condition and give a description of the attainable set. The equation includes a linear operator A defined in a Hilbert space H, in which by choosing H and A, we can obtain boundary controllability properties of numerous classes of nonlocal mixed value problems for wave equations which occur in a wide variety of physical systems. In this respect, we derived boundary controllability properties of the mixed problem for infinite many systems of wave equations, nonlocal mixed problem for degenerate wave equations and for high-order wave equations.