关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

The Koopman operator is a linear but infinite-dimensional operator that governs the evolution of scalar observables defined on the state space of an autonomous dynamical system and is a powerful tool for the analysis and decomposition of nonlinear dynamical systems In this manuscript, we present a data-driven method for approximating the leading eigenvalues, eigenfunctions, and modes of the Koopman operator The method requires a data set of snapshot pairs and a dictionary of scalar observables, but does not require explicit governing equations or interaction with a “black box” integrator We will show that this approach is, in effect, an extension of dynamic mode decomposition (DMD), which has been used to approximate the Koopman eigenvalues and modes Furthermore, if the data provided to the method are generated by a Markov process instead of a deterministic dynamical system, the algorithm approximates the eigenfunctions of the Kolmogorov backward equation, which could be considered as the “stochastic Koopman operator” (Mezic in Nonlinear Dynamics 41(1–3): 309–325, 2005) Finally, four illustrative examples are presented: two that highlight the quantitative performance of the method when presented with either deterministic or stochastic data and two that show potential applications of the Koopman eigenfunctions

A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition

In this paper we discuss two issues related to model reduction of deterministic or stochastic processes. The first is the relationship of the spectral properties of the dynamics on the attractor of the original, high-dimensional dynamical system with the properties and possibilities for model reduction. We review some elements of the spectral theory of dynamical systems. We apply this theory to obtain a decomposition of the process that utilizes spectral properties of the linear Koopman operator associated with the asymptotic dynamics on the attractor. This allows us to extract the almost periodic part of the evolving process. The remainder of the process has continuous spectrum. The second topic we discuss is that of model validation, where the original, possibly high-dimensional dynamics and the dynamics of the reduced model – that can be deterministic or stochastic – are compared in some norm. Using the “statistical Takens theorem” proven in (Mezic, I. and Banaszuk, A. Physica D, 2004) we argue that comparison of average energy contained in the finite-dimensional projection is one in the hierarchy of functionals of the field that need to be checked in order to assess the accuracy of the projection.

Spectral Properties of Dynamical Systems, Model Reduction and Decompositions

Brief paper: A state-feedback approach to event-based control

WITH the ever-widening scope of modern mathematical analysis and its many ramifications, it is quite impossible to include, in a single volume of reasonable size, an adequate and exhaustive discussion of the calculus in its more advanced stages. It therefore becomes necessary, in planning a thoroughly sound course in the subject, to consider several important aspects of the vast field confronting a modern writer. The limitation of space renders the selection of subject-matter fundamentally dependent upon the aim of the course, which may or may not be related to the content of specific examination syllabuses. Logical development, too, may lead to the inclusion of many topics which, at present, may only be of academic interest, while others, of greater practical value, may have to be omitted. The experience and training of the writer may also have, more or less, a bearing on both these considerations.Advanced CalculusBy Dr. C. A. Stewart. Pp. xviii + 523. (London: Methuen and Co., Ltd., 1940.) 25s.

Advanced Calculus I

We present efficient techniques for the numerical approximation of complicated dynamical behavior. In particular, we develop numerical methods which allow us to approximate Sinai--Ruelle--Bowen (SRB)-measures as well as (almost) cyclic behavior of a dynamical system. The methods are based on an appropriate discretization of the Frobenius--Perron operator, and two essentially different mathematical concepts are used: our idea is to combine classical convergence results for finite dimensional approximations of compact operators with results from ergodic theory concerning the approximation of SRB-measures by invariant measures of stochastically perturbed systems. The efficiency of the methods is illustrated by several numerical examples.

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On the Approximation of Complicated Dynamical Behavior

The computation of global invariant manifolds has seen renewed interest in recent years. We survey different approaches for computing a global stable or unstable manifold of a vector field, where we concentrate on the case of a two-dimensional manifold. All methods are illustrated with the same example — the two-dimensional stable manifold of the origin in the Lorenz system.

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A survey of methods for computing (un)stable manifolds of vector fields

This chapter focuses on set oriented numerical methods for dynamical systems. The set oriented numerical methods can be used to approximate different types of invariant sets or invariant manifolds but they also allow extracting statistical information on the dynamical behavior via the computation of natural invariant measures or almost invariant sets. In contrast to other numerical techniques, these methods do not rely on the computation of single long-term trajectories but rather use the information obtained from several short-term trajectories. Set oriented method can also be used for the computation of invariant manifolds. Although the method can, in principle, be applied to manifolds of arbitrary hyperbolic invariant sets. An important statistical characterization of the behavior of a dynamical system is given by so-called SRB (Sinai–Ruelle–Bowen) measures. The important property of these invariant measures is that they lend weight to a region in phase space according to the probability by which typical trajectories visit this region.

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Chapter 5 - Set Oriented Numerical Methods for Dynamical Systems

In a given dynamical system there are essentially two different types of information which could be of practical interest: on the one hand there is the need to describe the behavior of single trajectories in detail This information is helpful for the analysis of transient behavior and also in the investigation of geometric properties of dynamical systems On the other hand, if the underlying invariant set is generated by complicated dynamics then the computation of single trajectories may give misleading results In this case there still exists important set related information covering both topological and statistical aspects of the underlying dynamical behavior Within the DFG-Schwerpunkt we have focussed on the development of set oriented methods for the numerical approximation of 


invariant sets (eg invariant manifolds, global attractors, chain recurrent sets)


(natural) invariant measures


almost invariant sets

The Algorithms Behind GAIO — Set Oriented Numerical Methods for Dynamical Systems

The optimal control of a mechanical system is of crucial importance in many application areas. Typical examples are the determination of a time-minimal path in vehicle dynamics, a minimal energy trajectory in space mission design, or optimal motion sequences in robotics and biomechanics. In most cases, some sort of discretization of the original, infinite-dimensional optimization problem has to be performed in order to make the problem amenable to computations. The approach proposed in this paper is to directly discretize the variational description of the system's motion. The resulting optimization algorithm lets the discrete solution directly inherit characteristic structural properties from the continuous one like symmetries and integrals of the motion. We show that the DMOC (Discrete Mechanics and Optimal Control) approach is equivalent to a finite difference discretization of Hamilton's equations by a symplectic partitioned Runge-Kutta scheme and employ this fact in order to give a proof of convergence. The numerical performance of DMOC and its relationship to other existing optimal control methods are investigated.

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Oliver Junge

Papers

On the Approximation of Complicated Dynamical Behavior

A survey of methods for computing (un)stable manifolds of vector fields

Chapter 5 - Set Oriented Numerical Methods for Dynamical Systems

The Algorithms Behind GAIO — Set Oriented Numerical Methods for Dynamical Systems

Discrete mechanics and optimal control: an analysis ∗