  Contraction rates of time-varying maps induced by dynamical systems
illuminate a wide range of asymptotic properties with applications in stability
analysis and control theory. In finite-dimensional smoothly varying
inner-product spaces such as $\mathbb{R}^n$ and $\mathbb{C}^n$ with Riemannian
metrics, contraction rates can be estimated by upper-bounding the real
numerical range of the vector field's Jacobian. However, vector spaces with
norms other than $L^2$ commonly arise in the stability analysis of
infinite-dimensional systems such as those arising from partial differential
equations and continuum mechanics. To this end, we present a unified approach
to contraction analysis in Banach spaces using the theory of weighted
semi-inner products. We generalize contraction in a geodesic distance to
asymptotic stability of perturbations in smoothly varying semi-inner products,
and show that the latter is a dynamical invariant similar to the
coordinate-invariance of Lyapunov exponents. We show that contraction in
particular weighted spaces verifies asymptotic convergence to subspaces and
submanifolds, present applications to limit-cycle analysis and phase-locking
phenomena, and pose general conditions for inheritance of contraction
properties within coupled systems. We discuss contraction rates in Sobolev
spaces for retention of regularity in partial differential equations, and
suggest a type of weak solution defined by a vanishing contractive term.
Lastly, we present an application to machine learning, using weighted
semi-inner products to derive stability conditions for functional gradient
descent in a Banach space.
