Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Koopman invariant subspaces and finite linear representations of
  nonlinear dynamical systems for control   In this work, we explore finite-dimensional linear representations of
nonlinear dynamical systems by restricting the Koopman operator to an invariant
subspace. The Koopman operator is an infinite-dimensional linear operator that
evolves observable functions of the state-space of a dynamical system [Koopman
1931, PNAS]. Dominant terms in the Koopman expansion are typically computed
using dynamic mode decomposition (DMD). DMD uses linear measurements of the
state variables, and it has recently been shown that this may be too
restrictive for nonlinear systems [Williams et al. 2015, JNLS]. Choosing
nonlinear observable functions to form an invariant subspace where it is
possible to obtain linear models, especially those that are useful for control,
is an open challenge.
  Here, we investigate the choice of observable functions for Koopman analysis
that enable the use of optimal linear control techniques on nonlinear problems.
First, to include a cost on the state of the system, as in linear quadratic
regulator (LQR) control, it is helpful to include these states in the
observable subspace, as in DMD. However, we find that this is only possible
when there is a single isolated fixed point, as systems with multiple fixed
points or more complicated attractors are not globally topologically conjugate
to a finite-dimensional linear system, and cannot be represented by a
finite-dimensional linear Koopman subspace that includes the state. We then
present a data-driven strategy to identify relevant observable functions for
Koopman analysis using a new algorithm to determine terms in a dynamical system
by sparse regression of the data in a nonlinear function space [Brunton et al.
2015, arxiv]; we show how this algorithm is related to DMD. Finally, we
demonstrate how to design optimal control laws for nonlinear systems using
techniques from linear optimal control on Koopman invariant subspaces.

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The categories of this paper are: atom-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: nlin
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The categories of this paper are: ao-sci
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The categories of this paper are: econ
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The categories of this paper are: cond-mat
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The categories of this paper are: chem-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: bayes-an
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The categories of this paper are: hep-lat
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The categories of this paper are: adap-org
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The categories of this paper are: q-bio
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The categories of this paper are: quant-ph
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The categories of this paper are: stat
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The categories of this paper are: plasm-ph
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The categories of this paper are: mtrl-th