Tensor-Based Foundations for Analysis and Control of Complex Dynamical Systems with Higher-Order Interactions

Understanding how interacting components collectively influence the behavior of complex systems is a fundamental challenge across science and engineering. Such challenge arises in several natural and engineered systems, for example ecosystems, biological networks, autonomous robotic teams, social and technological networks, where the behavior of a group cannot be explained solely by interactions between pairs of components. Instead, multiple entities may interact simultaneously, giving rise to higher-order interactions that strongly influence system stability, resilience, adaptability, and collective behavior. Current computational methods often overlook these complex interactions, limiting the ability to accurately model, predict, and control large-scale systems. The research funded by this award will develop new mathematical and computational tools to improve our ability to understand, analyze and manage these complex interactions, advancing the scientific foundations needed for applications ranging from precision medicine to coordinated autonomous systems. The project will also train graduate and undergraduate students, develop open-source software and datasets, and disseminate research outcomes through workshops and educational outreach activities. This project develops a tensor-based framework for modeling, analyzing, and controlling higher-order interactions using homogeneous polynomial dynamical systems. The research integrates tensor theory, nonlinear systems theory, hypergraph theory, and data-driven methods to develop new approaches for system identification, model order reduction, stability analysis, controllability, stabilizability, and optimal control of higher-order dynamical systems. This framework exploits structured tensor representations and low-rank tensor decompositions to improve computational efficiency while enabling new theoretical insights into the dynamics of systems governed by higher-order interactions. The resulting mathematical and computational methods will be validated using both synthetic and real datasets and are expected to advance fundamental knowledge in dynamical systems, network science, optimization, and control while providing scalable tools for analyzing and controlling large-scale complex systems. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.