  This work establishes fundamental principles for verifying contract for
interconnected hybrid systems. When system's hybrid arcs conform to the
contract for a certain duration but subsequently violate it, the composition of
hybrid dynamical systems becomes challenging. The objective of this work is to
analyze the temporal satisfaction of the contract, allowing us to reason about
the compositions that do not violate the contract up to a certain point in a
hybrid time. Notions of weak and strong satisfaction of an assume-guarantee
contract are introduced. These semantics permits the compositional reasoning on
hybrid systems of varying complexity depending on the interconnection's type,
feedback or cascade. The results show that both semantics are compatible with
cascade composition, while strong semantic is required for feedback
composition. Moreover, we have shown how one can go from weak to strong
contract satisfaction. Finally, we have studied a particular class of hybrid
systems and we have shown that the concept of forward (pre-)invariant relative
to a contract makes it possible to deal with feedback compositions. These
results are demonstrated throughout the paper with simple numerical examples.
