A Clarke Matrix-Measure Contraction Certificate for Nonsmooth ODEs with Finite Spectral Verification and Mixed-Signal Circuit Applications

We study autonomous ordinary differential equations with locally Lipschitzvector fields that are not everywhere differentiable, as happens at switchingmanifolds and other non-smooth interfaces. For this setting, we introduce aClarke matrix measure obtained by taking the largest Euclidean matrix measureover the Clarke generalized Jacobian. On a forward-invariant set satisfyingthe segment condition for pairs of solutions, negativity of this measureyields strict Euclidean contraction with an explicit decay estimate. For continuouspiecewise-affine systems on convex domains, the same requirement isshown to be not merely sufficient but also necessary for strict Euclidean con-traction, and the verification reduces to a finite spectral test: one eigenvalueevaluation of the symmetric part of each mode matrix, with no decision variablesand no numerical solver. We further derive settling-time bounds anddemonstrate the analysis on two mixed-signal circuit models: a continuous-timesigma–delta modulator and a bang-bang phase-locked loop, illustratedthrough Monte Carlo trajectory studies and parameter sweeps of the certifiedrate. A numerical comparison with common quadratic Lyapunov methods,based on an exactly verifiable example, indicates the simplicity of the proposedtest and the conservatism it may entail.