abstract: "<jats:title>Abstract</jats:title>\n <jats:p>Landmark manifolds consist of finite collections of distinct points in an underlying space, referred to as landmark configurations. The dynamics of these configurations can be used to represent flows, such as solutions to ODEs or shape deformations. In this work, we consider landmark configurations in Euclidean space and study how they can be connected via flows of vector fields. For dimensions greater than or equal to two, we explicitly construct two vector fields whose flows can connect any pair of landmark configurations with the same cardinality. This property is known as exact universal interpolation. In dimension one, we show that the same result holds for pairs of landmark configurations that share the same relative ordering. In all dimensions, controllability is achieved using one constant vector field and one polynomial vector field of degree three.</jats:p>" author: Grong, Erlend and Vega-Molino, Sylvie author_list: - affiliation: [] family: Grong given: Erlend - affiliation: [] family: Vega-Molino given: Sylvie citations: - author: A Agrachev doi: 10.1051/cocv/2016029 first-page: '921' issue: '4' journal-title: ESAIM Control Optim Calc Var unstructured: Agrachev A, Baryshnikov Y, Sarychev A (2016) Ensemble controllability by Lie algebraic methods. ESAIM Control Optim Calc Var 22(4):921–938 volume: '22' year: '2016' - author: A Agrachev doi: 10.1007/s10883-021-09561-2 first-page: '989' issue: '4' journal-title: J Dyn Control Syst unstructured: Agrachev A, Sarychev A (2022) Control on the manifolds of mappings with a view to the deep learning. J Dyn Control Syst 28(4):989–1008 volume: '28' year: '2022' - unstructured: Agrachev AA, Caponigro M (2009) Controllability on the group of diffeomorphisms. 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Found Comput Math 12:295–325 volume: '12' year: '2012' doc_url: https://link.springer.com/content/pdf/10.1007/s00498-026-00435-1.pdf doi: 10.1007/s00498-026-00435-1 files: - s00498-026-00435-1.pdf issue: '2' journal: Mathematics of Control, Signals, and Systems language: en month: 6 pages: 275--289 papis_id: b3e254b5696b1e8e21ce3bacfff79973 publisher: Springer Science and Business Media LLC ref: Controllability_Grong_2026 time-added: 2026-08-19-22:34:42 title: Controllability on landmark manifolds for shapes and neural ODEs type: article url: https://doi.org/10.1007/s00498-026-00435-1 volume: '38' year: 2026