We consider operator-valued differential Lyapunov and Riccati equations,
where the operators B and C may be relatively unbounded with respect to A
(in the standard notation). In this setting, we prove that the singular values
of the solutions decay fast under certain conditions. In fact, the decay is
exponential in the negative square root if A generates an analytic semigroup
and the range of C has finite dimension. This extends previous similar
results for algebraic equations to the differential case. When the initial
condition is zero, we also show that the singular values converge to zero as
time goes to zero, with a certain rate that depends on the degree of
unboundedness of C. A fast decay of the singular values corresponds to a low
numerical rank, which is a critical feature in large-scale applications. The
results reported here provide a theoretical foundation for the observation
that, in practice, a low-rank factorization usually exists.Comment: Corrected some misconceptions, which lead to more general results
(e.g. exponential stability is no longer required). Also fixed some
off-by-one errors, improved the presentation, and added/extended several
remarks on possible generalizations. Now 22 pages, 8 figure