Solutions to Lyapunov stability problems of sets: Nonlinear systems with differentiable motions (Q1321856)

The author presents a new approach to the construction of a Lyapunov function for an autonomous nonlinear system (1) \(\dot x=f(x)\), \(x \in R^ n\), and to the exact determination of the domain of attraction of an asymptotically stable invariant set \(J\) of (1). The idea is to solve the equation \(\dot v=-p\) (or, alternatively, \(\dot w=-(1-w)p\), resulting from the substitution \(w(x)=1-\exp [-v(x)])\) with a suitably chosen right-hand side \(p=p(x)\) (satisfying \(p(x)=0\) if \(x \in J\), \(p(x)>0\) otherwise) along the trajectories of (1).