Estimates of the Kolmogorov n-width for nonlinear transformations with application to distributed-parameter control systems

Abstract

This paper aims at characterizing the approximability of bounded sets in the range of nonlinear operators in Banach spaces by finite-dimensional linear varieties. In particular, the class of operators we consider includes the endpoint maps of nonlinear distributed-parameter control systems. We describe the relationship between the Kolmogorov n-width of a bounded subset and the width of its image under an essentially nonlinear transformation. We propose explicit estimates of the n-width in the space of images in terms of the affine part of the corresponding operator and the width of its nonlinear perturbation. These $n$-width estimates enable us to describe the reachable sets for infinite-dimensional bilinear control systems, with applications to controlling the Euler-Bernoulli beam using a contraction force and to a single-input Schrödinger equation.

This is a preprint version of the paper submitted to the IEEE L-CSS and CDC 2024

References in corpus (1)

• Kolmogorov $n$-widths for linear dynamical systems