Multi-μ-stability and fixed-time multistability of switched fuzzy neural networks with discontinuous activation...

This paper addresses the multi-μ-stability and fixed-time multistability of switched fuzzy neural networks with discontinuous activation functions via an equilibrium-preconditioned controller. We first characterize the number, location and μ-stability of all equilibrium points for an n-neuron switched fuzzy neural network. Using these equilibrium data as prior information, we then design an equilibrium-preconditioned control law that achieves fixed-time convergence to one of the stable equilibria. By integrating state-space partition, differential-inclusion framework, and Lyapunov function approach, we derive explicit criteria ensuring that at most 9ⁿ equilibria coexist, among which 5ⁿ are locally μ-stable. The obtained criteria cover exponential and logarithmic stability as special cases of μ-stability. Once the equilibrium-preconditioned controller is activated, every trajectory converges to one of the 5ⁿ stable equilibria within a fixed time. A numerical example validates the theoretical counts and the fixed-time convergence performance.