  In this work, we propose a novel preconditioned Krylov subspace method for
solving an optimal control problem of wave equations, after explicitly
identifying the asymptotic spectral distribution of the involved sequence of
linear coefficient matrices from the optimal control problem. Namely, we first
show that the all-at-once system stemming from the wave control problem is
associated to a structured coefficient matrix-sequence possessing an eigenvalue
distribution. Then, based on such a spectral distribution of which the symbol
is explicitly identified, we develop an ideal preconditioner and two
parallel-in-time preconditioners for the saddle point system composed of two
block Toeplitz matrices. For the ideal preconditioner, we show that the
eigenvalues of the preconditioned matrix-sequence all belong to the set
$\left(-\frac{3}{2},-\frac{1}{2}\right)\bigcup
\left(\frac{1}{2},\frac{3}{2}\right)$ well separated from zero, leading to
mesh-independent convergence when the minimal residual method is employed. The
proposed {parallel-in-time} preconditioners can be implemented efficiently
using fast Fourier transforms or discrete sine transforms, and their
effectiveness is theoretically shown in the sense that the eigenvalues of the
preconditioned matrix-sequences are clustered around $\pm 1$, which leads to
rapid convergence. When these parallel-in-time preconditioners are not fast
diagonalizable, we further propose modified versions which can be efficiently
inverted. Several numerical examples are reported to verify our derived
localization and spectral distribution result and to support the effectiveness
of our proposed preconditioners and the related advantages with respect to the
relevant literature.
