My thesis

I worked on lifting techniques for dynamical systems. The methods I focused on were the Koopman operator and its adjoint, the Perron-Frobenius operator, and occupation measures. Those reformulate a complex non-linear dynamical system as a linear but infinite dimensional  dynamical system. This opens the door to apply techniques from linear functional analysis, notably convex optimization in the form of LPs and spectral theory. Nevertheless, the complexity of the original system is inherited in the shift from finite to infinite dimensional system. The course of dimension (for the underlying system) also transfers to the infinite dimensional case. I studied a certain type of sparsity to reduce the dimension of the original systems and its consequences for the infinite dimensional system. Below you find the "official" abstract.

Abstract:

In this thesis, we describe and analyze an interplay between dynamical systems, sparse structures, convex analysis, and functional analysis. We approach global attractors through an infinite dimensional linear programming problem (LP), investigate the Koopman and Perron-Frobenius semigroups of linear operators associated with a dynamical system, and show how a certain type of sparsity induces decompositions of several objects related to the dynamical systems; this includes the global attractor as well as the Koopman and Perron-Frobenius semigroups.
The first part of this work focuses on sparsity for dynamical systems. We define a notion of subsystems of a dynamical system and present how the system can be decomposed into its subsystems. This decomposition carries over to many important objects for the dynamical system, such as the maximum invariant set, the global attractor, or the stable manifold. We present the theoretical and practical limitations of our approach. Where those limitations do not apply, we show that sparsity can be exploited for computational tasks. One example is the computation of global attractors via the two infinite dimensional LPs that we propose. For polynomial dynamical systems, we solve these LPs in an established line of reasoning via techniques from polynomial optimization resulting in a sequence of semidefinite programs. This gives rise to a sequence of outer approximations of the global attractor which converges to the global attractor with respect to Lebesgue measure discrepancy.
For the Koopman and Perron-Frobenius semigroup, sparsity induces a certain block structure of these operators. This implies a decomposition of corresponding spectral objects such as eigenfunctions and invariant measures. A direct consequence is that subsystems induce eigenfunctions for the whole system and invariant measures for the dynamical system induce invariant measures of the subsystems. However, reversing this result is less straightforward. We show that for invariant measures this problem can be answered positively under necessary compatibility assumptions and for eigenfunctions we restrict to principal eigenfunctions and assume additional regularity.
We complement the sparse investigation of Koopman and Perron-Frobenius operators with their analysis on reproducing kernel Banach spaces (RKBS). This follows and extends a path of current research that investigates reproducing kernel Hilbert spaces (RKHS) as domains for Koopman and Perron-Frobenius operators. We provide a general framework for analysis of these operators on RKBS including their basic properties concerning closedness and boundedness. More precisely, we extend basic known properties of these operators from RKHSs to RKBSs and state new results, including symmetry and sparsity concepts, on these operators on RKBS for discrete and continuous time systems.