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.


  1. differential Riccati equations
  2. differential Lyapunov equations
  3. operator-valued
  4. infinite dimensional
  5. singular value decay
  6. low rank

MSC codes

  1. 47A62
  2. 47A11
  3. 49N10

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Information & Authors


Published In

cover image SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
Pages: 3598 - 3618
ISSN (online): 1095-7138


Submitted: 3 April 2018
Accepted: 31 July 2018
Published online: 9 October 2018


  1. differential Riccati equations
  2. differential Lyapunov equations
  3. operator-valued
  4. infinite dimensional
  5. singular value decay
  6. low rank

MSC codes

  1. 47A62
  2. 47A11
  3. 49N10



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