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. Third, we investigate the local convergence of these expansions. In particular, we exhibit arbitrarily small analytic vector fields for which the convergence of the Magnus expansion fails, even in very weak senses. Eventually, we derive approximate direct intrinsic representations for the state, particularly well designed for applications in control theory.

machine learningcommutative algebraalgebraic geometryalgebraic topologycombinatoricscategory theoryoperator algebrasrings and algebrasrepresentation theory

Audience: researchers in the topic

( chat | paper )

Algebraic and Combinatorial Perspectives in the Mathematical Sciences

Series comments: To receive announcements: Register into our mailing list by going to our main website www.math.ntnu.no/acpms/