Defense by Corbinian Schlosser

Date: April 26, 3 pm Paris time
Place: LAAS-CNRS, Toulouse, Salle de conférences
Title: Sparse structures and convex optimization for dynamical systems
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 and unstable manifold. We present practical use of our approach as well as its limitations from a theoretical and practical perspective.
One example where the computation benefits from a sparse decomposition is our proposed method for approximation of global attractors via the two infinite dimensional LPs. 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. For eigenfunctions we give a similar result for so-called principal eigenfunctions under an additional regularity assumption. 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.

The defense will be broadcasted on line at this Zoom link:
ID: 993 5086 4899
Code secret : 260423