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We consider a linear time-invariant system in discrete time where the stateand input signals satisfy a set of integral quadratic constraints (IQCs).Analogous to the autonomous linear systems case, we define a new notion ofspectral radius that exactly characterizes stability of this system. Inparticular, (i) when the spectral radius is less than one, we show that thesystem is asymptotically stable for all trajectories that satisfy the IQCs, and(ii) when the spectral radius is equal to one, we construct an unstabletrajectory that satisfies the IQCs. Furthermore, we connect our new definitionof the spectral radius to the existing literature on IQCs.

Integral Quadratic Constraints: Exact Convergence Rates and Worst-Case Trajectories