A stability criterion for nonlinear systems, derived by the first author (2000), can be viewed as a dual to Lyapunov's second theorem. The criterion is stated in terms of a function which can be interpreted as the stationary density of a substance that is generated all over the state space and flows along the system trajectories towards the equilibrium. The new criterion has a remarkable convexity property, which in this paper is used for controller synthesis via convex optimization. Numerical methods for verification of positivity of multivariate polynomials are used.