  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 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.
