Generalized Lyapunov approach for convergence of neural networks with discontinuous or non-Lipschitz activations

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A Comprehensive Review of Stability Analysis of Continuous-Time Recurrent Neural Networks (2014, background): “6) Discontinuous activation functions [68], [71], [72], [116], [173], [175], [183], [184], [216], [263]: Let g i (·) be a continuous nondecreasing function, and in every compact set of real space R, each g i (·) has only finite discontinuity points.”
Fixed/Preassigned-Time Synchronization of Complex Networks via Improving Fixed-Time Stability (2020, background): “Lemma 1 [34], [36]: Assume that V(y) is C-regular (regular, positive definite and radially unbounded) and y(t) is absolutely continuous on any compact subinterval of [0,+∞), then V(y(t)) and y(t) are differentiable for almost everywhere t ∈ [0,+∞) and dV(y(t)) dt = ζ T ẏ(t), for every ζ ∈ ∂V(y(t))”
Finite-time synchronization of complex networks with nonidentical discontinuous nodes (2013, background): “Lemma 1 [2] Suppose that function V (x) : R → R is C-regular, and that x(t) : [0,+∞) → R is absolutely continuous on any compact interval of [0,+∞).”