Behavior trees represent a hierarchical and modular way of combining several low-level control policies into a high-level task-switching policy. Hybrid dynamical systems can also be seen in terms of task switching between different policies, and therefore several comparisons between behavior trees and hybrid dynamical systems have been made, but only informally, and only in discrete time. A formal continuous-time formulation of behavior trees has been lacking. Additionally, convergence analyses

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... of specific classes of behavior tree designs have been made, but not for general designs. In this letter, we provide the first continuous-time formulation of behavior trees, show that they can be seen as discontinuous dynamical systems (a subclass of hybrid dynamical systems), which enables the application of existence and uniqueness results to behavior trees, and finally, provide sufficient conditions under which such systems will converge to a desired region of the state space for general designs. With these results, a large body of results on continuous-time dynamical systems can be brought to use when designing behavior tree controllers.