Dr. Matthew Kvalheim wins AFOSR Young Investigator Program (YIP) Award

Scope of the supported project

Dr. Matthew Kvalheim is the recipient of a Young Investigator Progam (YIP) Award from the Air Force Office of Scientific Research.  The award is to support a project titled A Topological Approach to Simplifying Hybrid Systems whose purpose is to develop mathematical techniques that simplify the analysis of hybrid systems.

Research

Many systems in engineering and everyday life combine continuous motion with sudden, discrete events. A walking robot, for example, moves continuously while each foot is in the air, but its behavior changes abruptly when a foot strikes the ground. Similar mathematical models arise in autonomous vehicles, mechanical systems with impacts, and devices whose controllers switch between different operating modes. These are called hybrid systems.

Hybrid systems can be extremely complicated, making it difficult to predict how they will behave or to design controllers that make them behave the way we want—for example, keeping a robot walking stably, steering an autonomous system toward a desired state, or preventing undesirable behavior.

The project's goals

This project develops new mathematical tools for replacing complicated hybrid-system models with simpler ones that preserve the behavior we care about. One approach is dimensionality reduction: finding a lower-dimensional description that captures the system's essential dynamics. Another is to find new coordinates in which the equations governing the system become simpler. Using ideas from topology and dynamical systems, we aim to build a general mathematical theory for carrying out these simplifications in hybrid systems. We will also investigate when such simplified representations can—or cannot—be discovered automatically from data by learning algorithms.

Training

The project also supports the training of the next generation of
mathematicians, as the award will support a Ph.D. student at UMBC.

Related work

• M. D. Kvalheim and P. Arathoon, Linearizability of flows by embeddings,  Selecta Mathematica, 32.2 (2026), 38. Journal article · Preprint
• M. D. Kvalheim and E. D. Sontag, Why should autoencoders work?, Transactions on Machine Learning Research (2024). Journal article · Preprint
• P. Arathoon and M. D. Kvalheim, Koopman embedding and super-linearization counterexamples with isolated equilibria(2023).  Preprint
• M. D. Kvalheim, P. Gustafson, and D. E. Koditschek, Conley’s fundamental theorem for a class of hybrid systems, SIAM Journal on Applied Dynamical systems, 20.2 (2021), pp. 784–825.Journal article · Preprint
• J. Eldering, M. D. Kvalheim, and S. Revzen, Global linearization and fiber bundle structure of invariant manifolds, Nonlinearity, 31.9 (2018), pp. 4202–4245. Journal article · Preprint
  

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