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In this work, we explore finite-dimensional linear representations ofnonlinear dynamical systems by restricting the Koopman operator to an invariantsubspace. The Koopman operator is an infinite-dimensional linear operator thatevolves observable functions of the state-space of a dynamical system [Koopman1931, PNAS]. Dominant terms in the Koopman expansion are typically computedusing dynamic mode decomposition (DMD). DMD uses linear measurements of thestate variables, and it has recently been shown that this may be toorestrictive for nonlinear systems [Williams et al. 2015, JNLS]. Choosingnonlinear observable functions to form an invariant subspace where it ispossible 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 analysisthat 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 quadraticregulator (LQR) control, it is helpful to include these states in theobservable subspace, as in DMD. However, we find that this is only possiblewhen there is a single isolated fixed point, as systems with multiple fixedpoints or more complicated attractors are not globally topologically conjugateto a finite-dimensional linear system, and cannot be represented by afinite-dimensional linear Koopman subspace that includes the state. We thenpresent a data-driven strategy to identify relevant observable functions forKoopman analysis using a new algorithm to determine terms in a dynamical systemby 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, wedemonstrate how to design optimal control laws for nonlinear systems usingtechniques from linear optimal control on Koopman invariant subspaces.

Koopman invariant subspaces and finite linear representations of nonlinear dynamical systems for control