Asymptotic stability of nonlinear systems

Assuming x¿ = f(x) is asymptotically stable with a known Lyapunov function, an expression is given for g(x); so that x¿ = f(x) + g(x) is also asymptotically stable. A similar result applies to the nonautonomous case. Applications are outlined to systems with control, and to the construction of stable systems.