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SUMMARY:Karine Beauchard (ENS Rennes)
DTSTART:20230414T130000Z
DTEND:20230414T140000Z
DTSTAMP:20260423T021603Z
UID:ACPMS/18
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/ACPMS/18/">O
 n expansions for nonlinear systems\, error estimates and convergence issue
 s</a>\nby Karine Beauchard (ENS Rennes) as part of Algebraic and Combinato
 rial Perspectives in the Mathematical Sciences\n\n\nAbstract\nExplicit for
 mulas 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 t
 he solution have been extensively studied.\nFirst\, we give a didactic rev
 iew of classical expansions for formal linear differential equations\, inc
 luding 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 introd
 uce a new mixed expansion\, designed to isolate the role of a time-invaria
 nt drift from the role of a time-varying perturbation.\nSecond\, in the co
 ntext of nonlinear ordinary differential equations driven by regular vecto
 r fields\, we give rigorous proofs of error estimates between the exact so
 lution and finite approximations of the formal expansions. In particular\,
  we derive new estimates focusing on the role of time-varying perturbation
 s.\nThird\, we investigate the local convergence of these expansions. In p
 articular\, we exhibit arbitrarily small analytic vector fields for which 
 the convergence of the Magnus expansion fails\, even in very weak senses.\
 nEventually\, we derive approximate direct intrinsic representations for t
 he state\, particularly well designed for applications in control theory.\
 n
LOCATION:https://researchseminars.org/talk/ACPMS/18/
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