On expansions for nonlinear systems, error estimates and convergence issues

Abstract

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.

arXiv commentUpdated numbering of theorems and equations to be the same as the published version

References in corpus (13)

• On Landau damping
• An efficient algorithm for computing the Baker-Campbell-Hausdorff series and some of its applications
• Splitting and composition methods in the numerical integration of differential equations
• Convergence of the Magnus series
• A Magnus- and Fer-type formula in dendriform algebras
• Faa di Bruno Hopf Algebra of the Output Feedback Group for Multivariable Fliess Operators
• The pre-Lie structure of the time-ordered exponential
• On the Baker-Campbell-Hausdorff Theorem: non-convergence and prolongation issues
• Generating Series for Networks of Chen-Fliess Series
• Necessary conditions for local controllability of a particular class of systems with two scalar controls
• On truncated logarithms of flows on a Riemannian manifold
• Growth of structure constants of free Lie algebras relative to Hall bases
• Butcher series: A story of rooted trees and numerical methods for evolution equations

Cited by in corpus (6)

• What is the Magnus Expansion?
• Growth of structure constants of free Lie algebras relative to Hall bases
• Small-Time Local Controllability of the multi-input bilinear Schrödinger equation thanks to a quadratic term
• Quadratic obstructions to small-time local controllability for multi-input systems
• A unified approach of obstructions to small-time local controllability for scalar-input systems
• Sub-Riemannian Landmark Matching and its interpretation as residual neural networks