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

Stillfjord, Tony LU orcid (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.

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Please use this url to cite or link to this publication:
author
publishing date
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}},
}