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

TL;DR

This paper proves that solutions to operator-valued differential Lyapunov and Riccati equations exhibit fast singular value decay under certain conditions, supporting low-rank approximations in large-scale control problems.

Contribution

It extends known singular value decay results from algebraic to differential operator equations with unbounded operators, providing theoretical insights for low-rank solutions.

Findings

Singular values decay exponentially in the negative square root when certain conditions hold.

Singular values converge to zero as time approaches zero for zero initial conditions.

The decay rate depends on the unboundedness of the operator C.

Abstract

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.