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Research

My research interests are in Dynamical Systems, Control Theory, and Applied Topology and Geometry.

Published or accepted journal papers:

  • Phase response curves and the role of coordinates.
    Simon Wilshin, Matthew D. Kvalheim, and Shai Revzen.
    (2024). Biological Cybernetics, Accepted (2024). (article, arXiv)

  • Why should autoencoders work?.
    Matthew D. Kvalheim and Eduardo D. Sontag.
    Transactions on Machine Learning Research (2024), pp. 1–24. (article, arXiv)

  • A compositional approach to certifying the almost global asymptotic stability of cascade systems.
    Jake Welde, Matthew D. Kvalheim, and Vijay Kumar.
    IEEE Control Systems Letters, 7 (2023), pp. 1969–1974. (article, arXiv)

  • Flux in tilted potential systems: negative resistance and persistence.
    Yuliy Baryshnikov and Matthew D. Kvalheim.
    Communications in Mathematical Physics, 400.2 (2023), pp. 853–930. (article, arXiv)

  • Obstructions to asymptotic stabilization.
    Matthew D. Kvalheim.
    SIAM Journal on Control and Optimization, 61.2 (2023), pp. 536–542. (article, arXiv)

  • A generalization of the Hopf degree theorem.
    Matthew D. Kvalheim.
    Proceedings of the American Mathematical Society, 151.1 (2023), pp. 453–454. (article, arXiv)

  • Planning of obstacle-aided navigation for multi-legged robots using a sampling-based method over directed graphs.
    Kaustav Chakraborty, Haodi Hu, Matthew D. Kvalheim, and Feifei Qian.
    IEEE Robotics and Automation Letters, 7.4 (2022), pp. 8861-8868. (article)

  • Necessary conditions for feedback stabilization and safety.
    Matthew D. Kvalheim and Daniel E. Koditschek.
    Journal of Geometric Mechanics, 14.4 (2022), pp. 659-693. (article, arXiv)

  • Existence and uniqueness of global Koopman eigenfunctions for stable fixed points and periodic orbits.
    Matthew D. Kvalheim and Shai Revzen.
    Physica D, 425 (2021), pp. 132959. (article, arXiv)

  • Families of periodic orbits: closed 1-forms and global continuability.
    Matthew D. Kvalheim and Anthony M. Bloch.
    Journal of Differential Equations, 285 (2021), pp. 211-257. (article, arXiv)

  • Conley’s fundamental theorem for a class of hybrid systems.
    Matthew D. Kvalheim, Paul Gustafson, and Daniel E. Koditschek.
    SIAM Journal on Applied Dynamical systems, 20.2 (2021), pp. 784-825. (article, arXiv)

  • Gait modeling and optimization for the perturbed Stokes regime.
    Matthew D. Kvalheim, Brian Bittner, and Shai Revzen.
    Nonlinear Dynamics, 97.4 (2019), pp. 2249-2270. (article, arXiv)

  • Global linearization and fiber bundle structure of invariant manifolds.
    Jaap Eldering, Matthew D. Kvalheim, and Shai Revzen.
    Nonlinearity, 31.9 (2018), pp. 4202-4245. (article, arXiv)

Published or accepted peer-reviewed conference papers:

  • The role of symmetry in constructing geometric flat outputs for free-flying robotic systems.
    Jake Welde, Matthew D. Kvalheim, and Vijay Kumar.
    IEEE International Conference on Robotics and Automation, Accepted (2023). (arXiv)

  • Generic properties of Koopman eigenfunctions for stable fixed points and periodic orbits.
    Matthew D. Kvalheim, David Hong, and Shai Revzen.
    IFAC-PapersOnline, 54.9 (2021), pp. 267-272. (article, arXiv)

  • Data-driven models of legged locomotion.
    Shai Revzen and Matthew D. Kvalheim.
    SPIE Defense + Security, International Society for Optics and Photonics (2015). (article)

Book chapters:

  • Templates and anchors.
    Matthew D. Kvalheim and Shai Revzen.
    Bioinspired legged locomotion, Ch. 3.2, Butterworth-Heinemann, Elsevier, Oxford (2017). (link)

  • Locomotion as an oscillator.
    Shai Revzen and Matthew D. Kvalheim.
    Bioinspired legged locomotion, Ch. 3.5, Butterworth-Heinemann, Elsevier, Oxford (2017). (link)

Papers under review and selected other preprints:

  • Relationships between necessary conditions for feedback stabilizability.
    Matthew D. Kvalheim.
    (2023). (arXiv)

  • Koopman embedding and super-linearization counterexamples with isolated equilibria.
    Philip Arathoon and Matthew D. Kvalheim.
    (2023). (arXiv)

  • Linearizability of flows by embeddings.
    Matthew D. Kvalheim and Philip Arathoon.
    (2023). (arXiv)

  • Estimating phase from observed trajectories using the temporal 1-form.
    Simon Wilshin, Matthew D. Kvalheim, Clayton Scott, and Shai Revzen.
    (2022). (arXiv)

  • A pasting lemma for Lipschitz functions.
    Matthew D. Kvalheim, Paul Gustafson, and Samuel A. Burden.
    (2021). (arXiv)

  • Poincaré-Hopf theorem for hybrid systems.
    Matthew D. Kvalheim.
    (2021). (arXiv)

PhD thesis:

Aspects of invariant manifold theory and applications, 207 pages, December 2018. (link)

Selected talks:

  • M. D. Kvalheim. “Linearizability of dynamical systems by embeddings.” Invited talk, Joint Mathematics Meetings (2024). San Francisco, CA, USA. (slides)
  • M. D. Kvalheim. “Discovering engineering (im)possibilities with geometry and topology.” Invited talk, University of Pennsylvania GRASP SFI Seminar (2023). Philadelphia, PA, USA. (video, slides)
  • M. D. Kvalheim. “Asymptotic stabilizability.” Invited talk, BIRS workshop Geometry, Topology and Control System Design (2023). Banff, AB, Canada. (slides)
  • M. D. Kvalheim. “Existence and uniqueness of Koopman eigenfunctions near stable equilibria and limit cycles.” Invited talk, International Symposium on Nonlinear Theory and Its Applications (NOLTA 2022). Online. (slides)
  • M. D. Kvalheim. “Large deviations, persistent homology, and Brownian conductors with negative resistance.” Invited talk, Cornell University Probability Seminar (2022). Online.
  • M. D. Kvalheim. “When can hybrid systems operate safely?” Invited talk, University of Michigan Control Seminar (2022). Online.
  • M. D. Kvalheim. “Flux in small noise dynamics: persistence and negative resistance.” Invited talk, Brown University, Division of Applied Mathematics (2022). Online.
  • S. Burden and M. D. Kvalheim. “When and how are hybrid dynamical systems conjugate to their classical quotients?” Invited talk, SIAM Conference on Control and its Applications (2021). Online.
  • M. D. Kvalheim. “Towards a Conley theory for hybrid dynamical systems.” Invited talk, Rutgers TRIPODS/DATA-INSPIRE Workshop on Dynamics, Topology, and Robotic Control (2021). Online. (video)
  • M. D. Kvalheim. “Hierarchical Composition via Collapse of Dimension in Dynamical Systems.” Invited talk, ETH Zürich Autonomy Talks (2021). Online.
  • A. M. Bloch and M. D. Kvalheim. “Geometry and dynamics of circulant systems.” Invited talk, AMS sectional meeting, special session on Geometric Dynamics (2020). Online.
  • M. D. Kvalheim, P. Gustafson, D. Koditschek. “Toward a task planning theory for robot hybrid dynamics.” Invited talk, BIRS-CMO workshop on Topological Complexity and Motion Planning (2020). Online.
  • M. D. Kvalheim, S. Wilshin, C. Scott, and S. Revzen. “Isochrons from short, noisy data. Invited talk, Dynamics Days Europe (2020). Online.
  • M. D. Kvalheim and S. Revzen. “Reduced-order models for locomotion in the perturbed Stokes regime.” Invited talk, SIAM Conference on Applications of Dynamical Systems (2019). Snowbird, UT.