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We show that the integral-constraint-based robust feedback stability theoremfor certain Lurye systems exhibits the property that the endogenousinput-output signals enjoy an exponential convergence rate for all initialconditions of the linear time-invariant subsystem. More generally, we provideconditions under which a feedback interconnection of possibly open-loopunbounded subsystems to admit such an exponential convergence property, usingperturbation analysis and a combination of tools including integral quadraticconstraints, directed gap measure, and exponential weightings. As anapplication, we apply the result to first-order convex optimisation methods. Inparticular, by making use of the Zames-Falb multipliers, we state conditionsfor these methods to converge exponentially when applied to strongly convexfunctions with Lipschitz gradients.

On the exponential convergence of input-output signals of nonlinear feedback systems