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SUMMARY:Stable infinite-dimensional dynamical systems - Anand Srinivasan
DTSTART:20221013T113000Z
DTEND:20221013T123000Z
UID:TALK184259@talks.cam.ac.uk
CONTACT:Raymond E. Goldstein
DESCRIPTION:Geodesic contraction in vector-valued differential equations i
 s readily verified by linearized operators which are uniformly negative-de
 finite in the Riemannian metric. In the infinite-dimensional setting\, how
 ever\, such analysis is generally restricted to norm-contracting systems. 
 We develop a generalization of geodesic contraction rates to Banach spaces
  using a smoothly-weighted semi-inner product structure on tangent spaces.
  We show that negative contraction rates in bijectively weighted spaces im
 ply asymptotic norm-contraction\, and apply recent results on asymptotic c
 ontractions in Banach spaces to establish the existence of fixed points. W
 e show that contraction in surjectively weighted spaces verify non-equilib
 rium asymptotic properties\, such as convergence to finite- and infinite-d
 imensional subspaces\, submanifolds\, limit cycles\, and phase-locking phe
 nomena. We use contraction rates in weighted Sobolev spaces to establish e
 xistence and continuous data dependence in nonlinear PDEs\, and pose a met
 hod for constructing weak solutions using vanishing one-sided Lipschitz ap
 proximations. We discuss applications to control and order reduction of PD
 Es.
LOCATION:MR15\,  Centre for Mathematical Sciences\, Wilberforce Road\, Cam
 bridge
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