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Explicit formulas expressing the solution to non-autonomous differential equations are of great importance in many application domains such as control theory or numerical operator splitting. In particular, intrinsic formulas allowing to decouple time-dependent features from geometry-dependent features of the solution have been extensively studied.
First, we give a didactic review of classical expansions for formal linear differential equations, including the celebrated Magnus expansion (associated with coordinates of the first kind) and Sussmann’s infinite product expansion (associated with coordinates of the second kind). Inspired by quantum mechanics, we introduce a new mixed expansion, designed to isolate the role of a time-invariant drift from the role of a time-varying perturbation.
Second, in the context of nonlinear ordinary differential equations driven by regular vector fields, we give rigorous proofs of error estimates between the exact solution and finite approximations of the formal expansions. In particular, we derive new estimates focusing on the role of time-varying perturbations. For scalar-input systems, we derive new estimates involving only a weak Sobolev norm of the input.
Third, we investigate the local convergence of these expansions. We recall known positive results for nilpotent dynamics and for linear dynamics. Nevertheless, we also exhibit arbitrarily small analytic vector fields for which the convergence of the Magnus expansion fails, even in very weak senses. We state an open problem concerning the convergence of Sussmann’s infinite product expansion.
Eventually, we derive approximate direct intrinsic representations for the state and discuss their link with the choice of an appropriate change of coordinates.
Beauchard, Karine 1 ; Le Borgne, Jérémy 1 ; Marbach, Frédéric 1
CC-BY 4.0
@article{CRMATH_2023__361_G1_97_0,
author = {Beauchard, Karine and Le Borgne, J\'er\'emy and Marbach, Fr\'ed\'eric},
title = {On expansions for nonlinear systems {Error} estimates and convergence issues},
journal = {Comptes Rendus. Math\'ematique},
pages = {97--189},
publisher = {Acad\'emie des sciences, Paris},
volume = {361},
number = {G1},
year = {2023},
doi = {10.5802/crmath.395},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.5802/crmath.395/}
}
TY - JOUR AU - Beauchard, Karine AU - Le Borgne, Jérémy AU - Marbach, Frédéric TI - On expansions for nonlinear systems Error estimates and convergence issues JO - Comptes Rendus. Mathématique PY - 2023 SP - 97 EP - 189 VL - 361 IS - G1 PB - Académie des sciences, Paris UR - http://geodesic.mathdoc.fr/articles/10.5802/crmath.395/ DO - 10.5802/crmath.395 LA - en ID - CRMATH_2023__361_G1_97_0 ER -
%0 Journal Article %A Beauchard, Karine %A Le Borgne, Jérémy %A Marbach, Frédéric %T On expansions for nonlinear systems Error estimates and convergence issues %J Comptes Rendus. Mathématique %D 2023 %P 97-189 %V 361 %N G1 %I Académie des sciences, Paris %U http://geodesic.mathdoc.fr/articles/10.5802/crmath.395/ %R 10.5802/crmath.395 %G en %F CRMATH_2023__361_G1_97_0
Beauchard, Karine; Le Borgne, Jérémy; Marbach, Frédéric. On expansions for nonlinear systems Error estimates and convergence issues. Comptes Rendus. Mathématique, Tome 361 (2023) no. G1, pp. 97-189. doi: 10.5802/crmath.395
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