Towards a Koopman theory for dynamical systems on completely regular spaces

Farkas, Bálint; Kreidler, Henrik

doi:10.1098/rsta.2019.0617

Cited by 5 publications

(8 citation statements)

References 39 publications

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“…The Koopman operator framework for infinite-dimensional systems is still in its infancy. Convergence properties of the finite-dimensional approximation of the Koopman operator should be thoroughly studied, in particular in light of the recent results by [6] in semigroup theory. This could provide some insight into the results obtained with the generalized EDMD method.…”

Section: Discussionmentioning

confidence: 99%

“…In fact, the infinitesimal generator of the Koopman semigroup cannot be defined unless the Koopman semigroup is strongly continuous (i.e., lim t↓0 K t ζ − ζ = 0 ∀ζ ∈ E ). This property does not hold in our setting, but can be satisfied with the mixed topology on C(U ), as shown in [6].…”

Section: Lie Generatormentioning

confidence: 93%

“…For this reason, strong convergence results can be obtained in [10] in the case of arbitrary bases, but a weaker result is obtained here (see Corollary 1). In this context, considering a space of bounded continuous functionals equipped with a mixed topology would allow the exploit of the strong continuity property of the semigroup, as shown in [6], and possibly recover stronger convergence results.…”

Section: Corollary 1 Under the Assumptions Of Proposition 1 But With ζmentioning

confidence: 99%

“…[5] followed a different path, defining the composition operator in the space of bounded continuous functionals and investigating the properties of the associated Lie generator. In recent work by [6], this approach was pushed further and leveraged in the framework of strongly continuous semigroups by considering a space of bounded continuous functionals equipped with a mixed topology. Finally, it is also recently that the Koopman operator framework has been considered for studying the spectral properties of nonlinear PDEs [7,8].…”

Section: Introductionmentioning

confidence: 99%

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Koopman Operator Framework for Spectral Analysis and Identification of Infinite-Dimensional Systems

Mauroy

2021

Mathematics

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We consider the Koopman operator theory in the context of nonlinear infinite-dimensional systems, where the operator is defined over a space of bounded continuous functionals. The properties of the Koopman semigroup are described and a finite-dimensional projection of the semigroup is proposed, which provides a linear finite-dimensional approximation of the underlying infinite-dimensional dynamics. This approximation is used to obtain spectral properties from the data, a method which can be seen as a generalization of the Extended Dynamic Mode Decomposition for infinite-dimensional systems. Finally, we exploit the proposed framework to identify (a finite-dimensional approximation of) the Lie generator associated with the Koopman semigroup. This approach yields a linear method for nonlinear PDE identification, which is complemented with theoretical convergence results.

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Section: Discussionmentioning

confidence: 99%

Section: Lie Generatormentioning

confidence: 93%

Section: Corollary 1 Under the Assumptions Of Proposition 1 But With ζmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Koopman Operator Framework for Spectral Analysis and Identification of Infinite-Dimensional Systems

Mauroy

2021

Mathematics

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“…In addition, also certain Koopman semigroups on C b (Ω), i.e. semigroups induced by a semiflow on a completely regular Hausdorff space Ω [21,32], require such a relaxed notion.…”

Section: Introductionmentioning

confidence: 99%

On equicontinuity and tightness of bi-continuous semigroups

Kruse,

Schwenninger

2021

Preprint

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In contrast to classical strongly continuous semigroups, the study of bi-continuous semigroups comes with some freedom in the properties of the associated locally convex topology. This paper aims to give minimal assumptions in order to recover typical features like tightness and equicontinuity with respect to the mixed topology as well as to carefully clarify on mutual relations between previously studied variants of these notions. The abstract results -exploiting techniques from topological vector spaces-are thoroughly discussed by means of several example classes, such as semigroups on spaces of bounded continuous functions.

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Convex semigroups on $$L^p$$-like spaces

2021

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In this paper, we investigate convex semigroups on Banach lattices with order continuous norm, having $$L^p$$ L p -spaces in mind as a typical application. We show that the basic results from linear $$C_0$$ C 0 -semigroup theory extend to the convex case. We prove that the generator of a convex $$C_0$$ C 0 -semigroup is closed and uniquely determines the semigroup whenever the domain is dense. Moreover, the domain of the generator is invariant under the semigroup, a result that leads to the well-posedness of the related Cauchy problem. In a last step, we provide conditions for the existence and strong continuity of semigroup envelopes for families of $$C_0$$ C 0 -semigroups. The results are discussed in several examples such as semilinear heat equations and nonlinear integro-differential equations.

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Towards a Koopman theory for dynamical systems on completely regular spaces

Cited by 5 publications

References 39 publications

Koopman Operator Framework for Spectral Analysis and Identification of Infinite-Dimensional Systems

Koopman Operator Framework for Spectral Analysis and Identification of Infinite-Dimensional Systems

On equicontinuity and tightness of bi-continuous semigroups

Convex semigroups on $$L^p$$-like spaces

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