Carleman linearization of nonlinear systems and its finite-section approximations

Arash Amini; Cong Zheng; Qiyu Sun; Nader Motee

doi:10.3934/dcdsb.2024102

Article Contents

2025, Volume 30, Issue 2: 577-603. Doi: 10.3934/dcdsb.2024102

This issue Previous Article Necessary conditions for similarity between two lattice differential equations Next Article Global dynamics of a Lotka-Volterra competition system in an advective patchy environment

Carleman linearization of nonlinear systems and its finite-section approximations

1.
Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX 78712, USA
2.
Department of Mathematics, University of Central Florida, Orlando, FL 32765, USA
3.
Mechanical Engineering and Mechanics Department, Lehigh University, Bethlehem, PA 18015, USA

Received: May 2024

Early access: August 2024

Published: February 2025

This work was supported in parts by the ONR N00014-23-1-2779.

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Abstract

The Carleman linearization is one of the mainstream approaches to lift a finite-dimensional nonlinear dynamical system into an infinite-dimensional linear system with the promise of providing accurate approximation of the original nonlinear system over larger regions around the equilibrium for longer time horizons with respect to the conventional first-order linearization approach. Finite-section approximations of the lifted system has been widely used to study dynamical and control properties of the original nonlinear system. In this context, some of the outstanding problems are to determine under what conditions, as the finite-section order (i.e., truncation length) increases, the trajectory of the resulting approximate linear system from the finite-section scheme converges to that of the original nonlinear system and whether the time interval over which the convergence happens can be quantified explicitly. In this paper, we provide explicit error bounds for the finite-section approximation and prove that the convergence is indeed exponential with respect to the finite-section order. For a class of stable nonlinear dynamical systems, it is shown that one can achieve exponential convergence over the entire time horizon up to infinity. Our results are practically plausible as our proposed error bound estimates can be used to compute proper truncation lengths for a given application, e.g., determining proper sampling period for model predictive control and reachability analysis for safety verifications. We validate our theoretical findings through several illustrative simulations.

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Mathematics Subject Classification: 34H05, 37M99, 65P99.

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Figure 1. Plotted are maximal approximation error $ E^\pm (x_0, N), 1\le N\le 100, 0\le x_0\le 1 $ in the logarithmic scale between the first block $ y_{1, N}^\pm $ of the finite section scheme (42) and the true solution $ x^\pm $ of the dynamical system (40) with positive sign (left) and negative sign (right) during the time period $ [0, T^*_\pm] $

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Figure 2. Plotted on the top are maximal approximation error $ E^+(x_0, N, T^*), 1\le N\le 100, 0\le x_0\le 1, $ in the logarithmic scale with $ T^* $ taking value $ 0.01 $ (top left) and $ 1 $ (top right) respectively. Plotted on the bottom are approximation error $ E^+(x_0, N)(t), 1\le N\le 100, 0\le t\le 1 $ in the logarithmic scale for the initial $ x_0 = 0.3 $ and $ 0.5 $ respectively

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Figure 3. Plotted on the top left is the vector field and limit cycle representation of the Van der Pol oscillator (44) with $ \mu = 0.5 $. On the top right, bottom left and bottom right are the maximal approximation error $ E(x_0,v_0,N,T^*) $ in the logarithmic scale, where $ T^* = 0.1, \mu = 0.5 $ and the truncation order $ N = 1, 10, 20 $ respectively

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