A Lyapunov-type function for generalized dynamical systems without uniqueness (Q1778222)

Generalized dynamic systems without uniqueness are considered. A function \(v(x)\) is constructed in the neighborhood of an asymptotically stable equilibrium. This function satisfies more weak conditions than Lyapunov's function (in general, \(v(x)\) can be discontinuous). The existence of such a function is a necessary and sufficient condition for asymptotic stability of the equilibrium. The results obtained are applicable to autonomous differential equations \[ \frac{dx}{dt}=f(x) \] with continuous \(f(x)\).