Contracting differential equations in weighted Banach spaces
Name
2204.05516v2.pdf
Description
Accepted version
Size
329.49 KB
Format
Adobe PDF
Checksum (MD5)
ab1e16db238de5894ada08b8b7bf3f00
Author(s)
Srinivasan, Anand
Slotine, Jean-Jacques
Date Issued
January 2023
Journal
Journal of Differential Equations
Publisher
Elsevier BV
Citation
Srinivasan, Anand and Slotine, Jean-Jacques. 2023. "Contracting differential equations in weighted Banach spaces." Journal of Differential Equations, 344.
Version
Author's final manuscript
Abstract
Geodesic contraction in vector-valued differential equations is readily verified by linearized operators which are uniformly negative-definite in the Riemannian metric. In the infinite-dimensional setting, however, 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 imply asymptotic norm-contraction, and apply recent results on asymptotic contractions in Banach spaces to establish the existence of fixed points. We show that contraction in surjectively weighted spaces verify non-equilibrium asymptotic properties, such as convergence to finite- and infinite-dimensional subspaces, submanifolds, limit cycles, and phase-locking phenomena. We use contraction rates in weighted Sobolev spaces to establish existence and continuous data dependence in nonlinear PDEs, and pose a method for constructing weak solutions using vanishing one-sided Lipschitz approximations. We discuss applications to control and order reduction of PDEs.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Massachusetts Institute of Technology. Nonlinear Systems Laboratory
Terms of Use
Creative Commons Attribution-Noncommercial-ShareAlike
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1016/j.jde.2022.10.040
