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Unified Large-Scale Theoretical and Computational Frameworks for Invariance and Composition of Open Hybrid Dynamical Systems

Hybrid systems are dynamical systems that exhibit both the continuous-time behaviors and the discrete transitions. Due to the intricate interplay between the continuous evolution and the instantaneous jumps, they show complex and rich dynamic characteristics that are challenging, or even formidable, to analyze especially with the conventional apparatus of the dynamic system theory.

This project is to establish both theoretical and computational frameworks to analyze and certify the intriguing behaviors of complicated open hybrid dynamical systems. In particular, the objective is to identify and construct the inherent structures of hybrid dynamics, such as topological properties and invariances, that can be preserved under the interaction with uncertain environments and composition over a complex network. This will be achieved with multidisciplinary collaborative efforts in dynamical systems and theoretical computer science, by synergistically integrating diverse approaches in geometry, topology, homology, category theory, stochastic analysis, and modern data-driven techniques. The novelty lies in establishing a trustworthy computational foundation that is carefully constructed in conjunction with the underlying geometry, leading to a significant generalization capacity and computation efficiency to handle non-trivial and non-conventional hybrid systems.