Keywords
Carleman linearization, dynamical systems with periodic vector fields, permutation-equivariant dynamical systems, finite-section approximation, explicit error bounds
Abstract
Carleman linearization has been widely employed in mathematical modeling and control theory and it could be used to investigate the stability of nonlinear systems, particularly in situations requiring higher-order accuracy. The essence of Carleman linearization is to lift a finite-dimensional nonlinear system to an infinite-dimensional linear system.
We consider nonlinear dynamical systems with periodic vector fields with multiple fundamental frequencies. We employ Fourier basis functions for Carleman-Fourier linearization, a method that transforms these systems into an infinite-dimensional linear model with an unbounded operator. This approach provides more accurate approximations than classical Carleman linearization, particularly in larger regions around the equilibrium and over extended time periods. For certain classes of systems, we demonstrate that exponential convergence is achievable across the entire time horizon. Our results are practically significant, as our proposed error bound estimates can guide in computing proper truncation lengths for various applications, such as determining the appropriate sampling period for model predictive control, conducting reachability analysis for safety verifications, and developing efficient algorithms for quantum computing.
To apply Carleman linearization to engineering applications, we must address both its accuracy and efficiency, as the latter poses a significant challenge. We have proposed a permutation-equivariant Carleman linearization, called PeCaL, to reduce the dimension of the finite-section approximation of Carleman Linearization when the nonlinear system is permutation-equivariant. We compare the time costs between classical Carleman Linearization and PeCaL with the same truncation order through simulations, alongside the explicit error bounds as previous works.
Completion Date
2024
Semester
Summer
Committee Chair
Sun, Qiyu
Degree
Doctor of Philosophy (Ph.D.)
College
College of Sciences
Department
Department of Mathematics
Degree Program
Doctoral program in Mathematics
Format
application/pdf
Identifier
DP0028888
URL
https://stars.library.ucf.edu/cgi/viewcontent.cgi?article=1484&context=etd2023
Language
English
Rights
In copyright
Release Date
8-15-2026
Length of Campus-only Access
1 year
Access Status
Doctoral Dissertation (Campus-only Access)
Campus Location
Orlando (Main) Campus
STARS Citation
Chen, Panpan, "Carleman Linearization for Nonlinear Systems and Their Explicit Error Bounds" (2024). Graduate Thesis and Dissertation 2023-2024. 474.
https://stars.library.ucf.edu/etd2023/474
Accessibility Status
Meets minimum standards for ETDs/HUTs
