Singular value decay of operator-valued differential Lyapunov and Riccati equations
(2018) In SIAM Journal on Control and Optimization 56(5). p.3598-3618- 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... (More)
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.
(Less)
- author
- Stillfjord, Tony LU
- publishing date
- 2018
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- Differential Lyapunov equations, Differential Riccati equations, Infinite dimensional, Low rank, Operator-valued, Singular value decay
- in
- SIAM Journal on Control and Optimization
- volume
- 56
- issue
- 5
- pages
- 21 pages
- publisher
- Society for Industrial and Applied Mathematics
- external identifiers
-
- scopus:85056129700
- ISSN
- 0363-0129
- DOI
- 10.1137/18M1178815
- language
- English
- LU publication?
- no
- additional info
- Publisher Copyright: © 2018 Society for Industrial and Applied Mathematics
- id
- 6c8584a2-d23c-4d3d-b58c-76c7276faaea
- date added to LUP
- 2024-01-23 17:38:09
- date last changed
- 2024-02-26 08:58:00
@article{6c8584a2-d23c-4d3d-b58c-76c7276faaea, abstract = {{<p>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.</p>}}, author = {{Stillfjord, Tony}}, issn = {{0363-0129}}, keywords = {{Differential Lyapunov equations; Differential Riccati equations; Infinite dimensional; Low rank; Operator-valued; Singular value decay}}, language = {{eng}}, number = {{5}}, pages = {{3598--3618}}, publisher = {{Society for Industrial and Applied Mathematics}}, series = {{SIAM Journal on Control and Optimization}}, title = {{Singular value decay of operator-valued differential Lyapunov and Riccati equations}}, url = {{http://dx.doi.org/10.1137/18M1178815}}, doi = {{10.1137/18M1178815}}, volume = {{56}}, year = {{2018}}, }