  We focus on the problem of optimal control of large-scale systems whose
models are obtained by discretization of partial differential equations using
the Finite Element (FE) or Finite Difference (FD) methods. The motivation for
studying this pressing problem originates from the fact that the classical
numerical tools used to solve low-dimensional optimal control problems are
computationally infeasible for large-scale systems. Furthermore, although the
matrices of large-scale FE or FD models are usually sparse banded or highly
structured, the optimal control solution computed using the classical methods
is dense and unstructured. Consequently, it is not suitable for efficient
centralized and distributed real-time implementations. We show that the a
priori (sparsity) patterns of the exact solutions of the generalized Lyapunov
equations for FE and FD models are banded matrices. The a priori pattern
predicts the dominant non-zero entries of the exact solution. We furthermore
show that for well-conditioned problems, the a priori patterns are not only
banded but also sparse matrices. On the basis of these results, we develop two
computationally efficient methods for computing sparse approximate solutions of
generalized Lyapunov equations. Using these two methods and the inexact Newton
method, we show that the solution of the generalized Riccati equation can be
approximated by a banded matrix. This enables us to develop a novel
computationally efficient optimal control approach that is able to preserve the
sparsity of the control law. We perform extensive numerical experiments that
demonstrate the effectiveness of our approach.
