Sparse decompositions of nonlinear dynamical systems and applications to moment-sum-of-squares relaxations

Sparse decompositions of nonlinear dynamical systems and applications to moment-sum-of-squares relaxations | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Sparse decompositions of nonlinear dynamical systems and applications to moment-sum-of-squares relaxations Corbinian Schlosser, Milan Korda This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6513566/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 10 You are reading this latest preprint version Abstract We propose a general sparse decomposition of dynamical systems provided that the vector field and constraint set possess certain sparse structures, which we call subsystems. This notion is based on causal dependence in the dynamics between the different states. It gives rise to sparse descriptions for fundamental problems from nonlinear dynamical systems: The characterization of the region of attraction, the maximum positively invariant set, and the global attractor. The decompositions can be paired with any method for computing (outer) approximations of these sets, and reduce the task to lower dimensional systems. We illustrate this by methods from previous work based on infinite-dimensional linear programming. This exhibits one example where the curse of dimensionality is present and hence dimension reduction is crucial. In this context, for polynomial dynamics, we show that these problems admit convergent sparse sum-of-squares (SOS) approximations. We illustrate the computational benefit with numerical toy examples. nonlinear control large-scale optimization sparsity decomposition network control sum-of-squares Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 15 Jul, 2025 Reviews received at journal 01 Jul, 2025 Reviews received at journal 27 May, 2025 Reviewers agreed at journal 05 May, 2025 Reviewers agreed at journal 02 May, 2025 Reviewers agreed at journal 02 May, 2025 Reviewers invited by journal 02 May, 2025 Editor assigned by journal 25 Apr, 2025 Submission checks completed at journal 24 Apr, 2025 First submitted to journal 23 Apr, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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