Smooth blending of nonlinear controllers using density functions

A new convergence criterion for nonlinear systems was recently derived by the first author. The criterion is similar to Lyapunov\´s second theorem but differs in several respects. In particular, it has a remarkable convexity property in the context of control synthesis. While the set of control Lyapunov functions for a given system may not even be connected, the corresponding set of "density" functions is always convex. In this paper, we will demonstrate how this opens new possibilities in synthesis of nonlinear controllers. We also give an alternative version of the criterion, that proves attractivity of a set rather than a point in the state space.