The controllability problem for abstract wave equations and its applications

Abstract

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 bound- ary 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.