Duration Calculus is a real-time interval logic which can be used to
specify and reason about timing and logical constraints on discrete
states in a dynamic system.  It has been used to specify and verify
designs for a number of real-time systems.  This paper extends the
Duration Calculus with notations to capture properties of piecewise
continuous states.  This is useful for reasoning about hybrid systems
with a mixture of continuous and discrete states.  The proof theory of
Duration Calculus is extended such that results proven using
mathematical analysis can be used freely in the logic.  This provides a
flexible interface to conventional control theory.

{\bf Keywords:}Requirements, software design, hybrid systems,
real-time systems, control theory, interval logic, durations.