Singular value decay of operator-valued differential Lyapunov and Riccati equations
Provides theoretical justification for low-rank numerical methods in large-scale control and filtering problems.
The paper proves that singular values of solutions to operator-valued differential Lyapunov and Riccati equations decay exponentially under certain conditions, extending results from algebraic to differential equations. This provides a theoretical basis for low-rank approximations in large-scale applications.
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.