Linearization in the large of nonlinear systems and Koopman operator spectrum

Lan, Yueheng; Mezić, Igor

doi:10.1016/j.physd.2012.08.017

Cited by 166 publications

(156 citation statements)

References 20 publications

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“…Any subset of eigenfunctions of the Koopman operator form an invariant subspace, by definition. However, these eigenfunctions may be arbitrarily complex, as they are intricately related to the phase space geometry of the dynamical system [120][121][122]. Similar to DMD, we would like the Koopman-invariant subspaces to include the state variables x directly as observables.…”

Section: Method: Koopman Observable Subspaces Containing the Statementioning

confidence: 99%

Including inputs and control within equation-free architectures for complex systems

Proctor

Brunton

Kutz

2016

Eur. Phys. J. Spec. Top.

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Abstract. The increasing ubiquity of complex systems that require control is a challenge for existing methodologies in characterization and controller design when the system is high-dimensional, nonlinear, and without physics-based governing equations. We review standard model reduction techniques such as Proper Orthogonal Decomposition (POD) with Galerkin projection and Balanced POD (BPOD). Further, we discuss the link between these equation-based methods and recently developed equation-free methods such as the Dynamic Mode Decomposition and Koopman operator theory. These data-driven methods can mitigate the challenge of not having a well-characterized set of governing equations. We illustrate that this equation-free approach that is being applied to measurement data from complex systems can be extended to include inputs and control. Three specific research examples are presented that extend current equation-free architectures toward the characterization and control of complex systems. These examples motivate a potentially revolutionary shift in the characterization of complex systems and subsequent design of objective-based controllers for data-driven models.

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Section: Method: Koopman Observable Subspaces Containing the Statementioning

confidence: 99%

Including inputs and control within equation-free architectures for complex systems

Proctor

Brunton

Kutz

2016

Eur. Phys. J. Spec. Top.

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“…They are either empty with most of the types of attractors (fixed point, limit cycle, quasiperiodic tori, see e.g. [3], [4], [6], [9]) in well-chosen spaces of observables, or they correspond to the asymptotic ergodic dynamics on (strange) attractors, therefore carrying no information on stability.…”

Section: Spectral Analysismentioning

confidence: 99%

“…For instance, they define new coordinates z i = φ λ * i (x) associated with linear dynamicsż i = λ * i z i [6] and are therefore related to the rate of convergence of the trajectories. In particular, the level sets of the eigenfunction φ λ * 1 correspond to the notion of isostables defined in [9]: they are the sets of points that converge "synchronously" toward the fixed point.…”

Section: ) Lyapunov Functions and Contracting Metricsmentioning

confidence: 99%

“…In particular, we focus on the spectral properties of the Koopman operator and we establish a relationship between global stability and the existence of specific eigenfunctions. Beyond their relationship to stability, the eigenfunctions are related to important geometric properties of the dynamics (derivation of special Lyapunov functions and contracting metrics, global linearization [6], isostables [9], etc.). This analysis mirrors the spectral approach to stability in linear systems.…”

Section: Introductionmentioning

confidence: 99%

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A spectral operator-theoretic framework for global stability

Mauroy

Mezić

2013

52nd IEEE Conference on Decision and Control

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Abstract-The global description of a nonlinear system through the linear Koopman operator leads to an efficient approach to global stability analysis. In the context of stability analysis, not much attention has been paid to the use of spectral properties of the operator. This paper provides new results on the relationship between the global stability properties of the system and the spectral properties of the Koopman operator. In particular, the results show that specific eigenfunctions capture the system stability and can be used to recover known notions of classical stability theory (e.g. Lyapunov functions, contracting metrics). Finally, a numerical method is proposed for the global stability analysis of a fixed point and is illustrated with several examples.

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“…С формальной точки зрения, DMD служит методом построения аппроксимации оператора Купмана [12,13], который является линейным оператором, описывающим динамику системы в некотором пространстве наблюдаемых (нелинейных функций динамических переменных). Оператор Купмана тесно связан (сопряжен) с оператором Перрона-Фробениуса (пропагатором обобщенного уравнения Лиувилля), описывающим линейную динамику плотности вероятности [14][15][16][17][18][19].…”

Section: Introductionunclassified

On the application of the dynamic mode decomposition in problems of computational fluid dynamics

Alekseev¹,

Bondarev

2018

KIAM Prepr.

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Linearization in the large of nonlinear systems and Koopman operator spectrum

Cited by 166 publications

References 20 publications

Including inputs and control within equation-free architectures for complex systems

Including inputs and control within equation-free architectures for complex systems

A spectral operator-theoretic framework for global stability

On the application of the dynamic mode decomposition in problems of computational fluid dynamics

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