  In the present paper, we consider large-scale differential Lyapunov matrix
equations having a low rank constant term. We present two new approaches for
the numerical resolution of such differential matrix equations. The first
approach is based on the integral expression of the exact solution and an
approximation method for the computation of the exponential of a matrix times a
block of vectors. In the second approach, we first project the initial problem
onto a block (or extended block) Krylov subspace and get a low-dimensional
differential Lyapunov matrix equation. The latter differential matrix problem
is then solved by the Backward Differentiation Formula method (BDF) and the
obtained solution is used to build the low rank approximate solution of the
original problem. The process being repeated until some prescribed accuracy is
achieved. We give some new theoretical results and present some numerical
experiments.
