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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 valuesof the solutions decay fast under certain conditions. In fact, the decay isexponential in the negative square root if $A$ generates an analytic semigroupand the range of $C$ has finite dimension. This extends previous similarresults for algebraic equations to the differential case. When the initialcondition is zero, we also show that the singular values converge to zero astime goes to zero, with a certain rate that depends on the degree ofunboundedness of $C$. A fast decay of the singular values corresponds to a lownumerical rank, which is a critical feature in large-scale applications. Theresults reported here provide a theoretical foundation for the observationthat, in practice, a low-rank factorization usually exists.

Singular value decay of operator-valued differential Lyapunov and Riccati equations