On expansions for nonlinear systems Error estimates and convergence issues
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 featur...
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| Format: | Article |
| Language: | English |
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Académie des sciences
2023-01-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.395/ |
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| author | Beauchard, Karine Le Borgne, Jérémy Marbach, Frédéric |
| author_facet | Beauchard, Karine Le Borgne, Jérémy Marbach, Frédéric |
| author_sort | Beauchard, Karine |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-79d0059fab714472a848685641667f8d |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2023-01-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-79d0059fab714472a848685641667f8d2025-02-07T11:06:07ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692023-01-01361G19718910.5802/crmath.39510.5802/crmath.395On expansions for nonlinear systems Error estimates and convergence issuesBeauchard, Karine0Le Borgne, Jérémy1Marbach, Frédéric2Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, FranceUniv Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, FranceUniv Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, FranceExplicit 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.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.395/ |
| spellingShingle | Beauchard, Karine Le Borgne, Jérémy Marbach, Frédéric On expansions for nonlinear systems Error estimates and convergence issues Comptes Rendus. Mathématique |
| title | On expansions for nonlinear systems Error estimates and convergence issues |
| title_full | On expansions for nonlinear systems Error estimates and convergence issues |
| title_fullStr | On expansions for nonlinear systems Error estimates and convergence issues |
| title_full_unstemmed | On expansions for nonlinear systems Error estimates and convergence issues |
| title_short | On expansions for nonlinear systems Error estimates and convergence issues |
| title_sort | on expansions for nonlinear systems error estimates and convergence issues |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.395/ |
| work_keys_str_mv | AT beauchardkarine onexpansionsfornonlinearsystemserrorestimatesandconvergenceissues AT leborgnejeremy onexpansionsfornonlinearsystemserrorestimatesandconvergenceissues AT marbachfrederic onexpansionsfornonlinearsystemserrorestimatesandconvergenceissues |