The Controllability Problem for Abstract Wave Equations and Its Applications

Abstract

In this paper, the authors study the boundary controllability problem of a wave equation with a nonlocal term and a linear operator A defined in a Hilbert space H, where A is a sectorial operator on H with discrete spectrum and corresponding eigenfunctions is a orthonormal system in H. Based on theory of Riesz basis, they give a Kalman-type condition and a description of the attainable set. With these results, by choosing H and A, they can obtain boundary controllability properties of numerous classes of nonlocal mixed value problems for wave equations. In other words, they obtained boundary controllability of the mixed problem for wave equations, nonlocal mixed problem for degenerate wave equations and for high-order wave equations. Since the key assumption on the discrete spectrum, it seems that the method in this paper cannot be employed to study controllability problem of multidimensional wave equations.