Convergence Analysis of Lie and Strang Splitting for Operator-Valued Differential Riccati Equations
Hansen, Eskil LU
; Stillfjord, Tony
LU
and Åberg, Teodor
LU
(2026)
In SIAM Journal on Numerical Analysis
64(3).
p.957-988
- Abstract
- Differential Riccati equations (DREs) are semilinear matrix- or operator-valued differential equations with quadratic nonlinearities. They arise in many different areas, and are particularly important in optimal control of linear quadratic regulators, where they provide the optimal feedback control laws. In the context of control of partial differential equations, these Riccati equations are operator-valued. To approximate their solutions, both spatial and temporal discretizations are needed. While the former have been well analyzed in the literature, there are very few rigorous convergence analyses of time stepping methods applied to DREs, particularly in the infinite-dimensional, operator-valued setting. In view of this, we analyze two... (More)
- Differential Riccati equations (DREs) are semilinear matrix- or operator-valued differential equations with quadratic nonlinearities. They arise in many different areas, and are particularly important in optimal control of linear quadratic regulators, where they provide the optimal feedback control laws. In the context of control of partial differential equations, these Riccati equations are operator-valued. To approximate their solutions, both spatial and temporal discretizations are needed. While the former have been well analyzed in the literature, there are very few rigorous convergence analyses of time stepping methods applied to DREs, particularly in the infinite-dimensional, operator-valued setting. In view of this, we analyze two numerical time-stepping schemes, the Lie and Strang splitting methods, in such a setting. The analysis relies on the assumption that the uncontrolled system is parabolic and that either the initial condition is sufficiently smooth, or the nonlinearity in the DRE is sufficiently smoothing. These assumptions are mild, in the sense that they are not enough to even guarantee continuity in the operator norm of the exact solution to the DRE. However, they imply certain regularity in a pointwise sense, which can be leveraged to prove convergence in the operator norm with the classical orders. The results are illustrated by four numerical experiments, where convergence with the expected order is correlated with the relevant assumptions being fulfilled. The experiments also demonstrate that matrix-valued DREs which arise as spatial discretizations of operator-valued DREs behave similarly, unless the discretization is coarse. (Less)
- Abstract (Swedish)
- Differential Riccati equations (DREs) are semilinear matrix- or operator-valued differential equations with quadratic nonlinearities. They arise in many different areas, and are particularly important in optimal control of linear quadratic regulators, where they provide the optimal feedback control laws. In the context of control of partial differential equations, these Riccati equations are operator-valued. To approximate their solutions, both spatial and temporal discretizations are needed. While the former have been well analyzed in the literature, there are very few rigorous convergence analyses of time stepping methods applied to DREs, particularly in the infinite-dimensional, operator-valued setting. In view of this, we analyze two... (More)
- Differential Riccati equations (DREs) are semilinear matrix- or operator-valued differential equations with quadratic nonlinearities. They arise in many different areas, and are particularly important in optimal control of linear quadratic regulators, where they provide the optimal feedback control laws. In the context of control of partial differential equations, these Riccati equations are operator-valued. To approximate their solutions, both spatial and temporal discretizations are needed. While the former have been well analyzed in the literature, there are very few rigorous convergence analyses of time stepping methods applied to DREs, particularly in the infinite-dimensional, operator-valued setting. In view of this, we analyze two numerical time-stepping schemes, the Lie and Strang splitting methods, in such a setting. The analysis relies on the assumption that the uncontrolled system is parabolic and that either the initial condition is sufficiently smooth, or the nonlinearity in the DRE is sufficiently smoothing. These assumptions are mild, in the sense that they are not enough to even guarantee continuity in the operator norm of the exact solution to the DRE. However, they imply certain regularity in a pointwise sense, which can be leveraged to prove convergence in the operator norm with the classical orders. The results are illustrated by four numerical experiments, where convergence with the expected order is correlated with the relevant assumptions being fulfilled. The experiments also demonstrate that matrix-valued DREs which arise as spatial discretizations of operator-valued DREs behave similarly, unless the discretization is coarse. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/ec8a493f-060f-4558-892c-cacb178f3ad0
- author
- Hansen, Eskil
LU
; Stillfjord, Tony
LU
and Åberg, Teodor
LU
- organization
- publishing date
- 2026-06-23
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- differentiella Riccati-ekvationer, operator-värda, splitting-metoder, konvergensanalys, differential Riccati equations, operator-valued, splitting schemes, convergence analysis
- in
- SIAM Journal on Numerical Analysis
- volume
- 64
- issue
- 3
- pages
- 32 pages
- publisher
- Society for Industrial and Applied Mathematics
- external identifiers
-
- scopus:105044559408
- ISSN
- 0036-1429
- DOI
- 10.1137/25M1754406
- project
- Numerical methods for differential Riccati equations
- Moving domain decomposition methods for parabolic PDEs
- language
- English
- LU publication?
- yes
- id
- ec8a493f-060f-4558-892c-cacb178f3ad0
- alternative location
- https://doi.org/10.1137/25M1754406
- date added to LUP
- 2026-06-23 11:51:17
- date last changed
- 2026-09-08 13:10:19
@article{ec8a493f-060f-4558-892c-cacb178f3ad0,
abstract = {{Differential Riccati equations (DREs) are semilinear matrix- or operator-valued differential equations with quadratic nonlinearities. They arise in many different areas, and are particularly important in optimal control of linear quadratic regulators, where they provide the optimal feedback control laws. In the context of control of partial differential equations, these Riccati equations are operator-valued. To approximate their solutions, both spatial and temporal discretizations are needed. While the former have been well analyzed in the literature, there are very few rigorous convergence analyses of time stepping methods applied to DREs, particularly in the infinite-dimensional, operator-valued setting. In view of this, we analyze two numerical time-stepping schemes, the Lie and Strang splitting methods, in such a setting. The analysis relies on the assumption that the uncontrolled system is parabolic and that either the initial condition is sufficiently smooth, or the nonlinearity in the DRE is sufficiently smoothing. These assumptions are mild, in the sense that they are not enough to even guarantee continuity in the operator norm of the exact solution to the DRE. However, they imply certain regularity in a pointwise sense, which can be leveraged to prove convergence in the operator norm with the classical orders. The results are illustrated by four numerical experiments, where convergence with the expected order is correlated with the relevant assumptions being fulfilled. The experiments also demonstrate that matrix-valued DREs which arise as spatial discretizations of operator-valued DREs behave similarly, unless the discretization is coarse.}},
author = {{Hansen, Eskil and Stillfjord, Tony and Åberg, Teodor}},
issn = {{0036-1429}},
keywords = {{differentiella Riccati-ekvationer; operator-värda; splitting-metoder; konvergensanalys; differential Riccati equations; operator-valued; splitting schemes; convergence analysis}},
language = {{eng}},
month = {{06}},
number = {{3}},
pages = {{957--988}},
publisher = {{Society for Industrial and Applied Mathematics}},
series = {{SIAM Journal on Numerical Analysis}},
title = {{Convergence Analysis of Lie and Strang Splitting for Operator-Valued Differential Riccati Equations}},
url = {{https://lup.lub.lu.se/search/files/253632546/HansenStillfjordAaberg2026.pdf}},
doi = {{10.1137/25M1754406}},
volume = {{64}},
year = {{2026}},
}