{"ID":6024168,"CreatedAt":"2026-07-08T01:00:23.257252134Z","UpdatedAt":"2026-07-09T22:59:56.587686154Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.05687","arxiv_id":"2607.05687","title":"Sphere Constraints and Harmonic Map Flow: Controllability and Reachability by Low-Mode Forcing","abstract":"We study the controllability and reachability of sphere-constrained evolution equations under degenerate (low-mode) forcing, with the harmonic map heat flow as the principal application. Exploiting the underlying geometric structure, we reformulate the problem as an infinite-dimensional control-affine system in Fourier variables and analyze the Lie algebra generated by the controlled vector fields. We prove that iterated Lie brackets generate new admissible directions, providing a mechanism through which finitely many control modes propagate their influence across infinitely many Fourier components. The results provide a Lie-algebraic framework for controlling manifold-valued evolution equations.","short_abstract":"We study the controllability and reachability of sphere-constrained evolution equations under degenerate (low-mode) forcing, with the harmonic map heat flow as the principal application. Exploiting the underlying geometric structure, we reformulate the problem as an infinite-dimensional control-affine system in Fourier...","url_abs":"https://arxiv.org/abs/2607.05687","url_pdf":"https://arxiv.org/pdf/2607.05687v1","authors":"[\"Debopriya Mukherjee\",\"Kistosil Fahim\",\"Erika Hausenblas\"]","published":"2026-07-06T23:12:30Z","proceeding":"math.OC","tasks":"[\"math.OC\",\"math.AP\"]","methods":"[]","has_code":false}