  This work extends reachability analyses based on ellipsoidal techniques to
Linear Time Invariant (LTI) systems subject to an integral quadratic constraint
(IQC) between the past state and disturbance signals , interpreted as an
input-output energetic constraint. To compute the reachable set, the LTI system
is augmented with a state corresponding to the amount of energy still available
before the constraint is violated. For a given parabolic set of initial states,
the reachable set of the augmented system is overapproximated with a
time-varying parabolic set. Parameters of this paraboloid are expressed as the
solution of an Initial Value Problem (IVP) and the overapproximation
relationship with the reachable set is proved. This paraboloid is actually
supported by the reachable set on so-called touching trajectories. Finally , we
describe a method to generate all the supporting paraboloids and prove that
their intersection is an exact characterization of the reachable set. This work
provides new practical means to compute overapproximation of reachable sets for
a wide variety of systems such as delayed systems, rate limiters or
energy-bounded linear systems.
