Nonlinear control synthesis for a class of polynomial systems based on density function

For polynomial nonlinear systems with the origin as a fixed point, this paper presents a novel asymptotical stability criterion that the existence of a positive density function with its time derivative greater than zero implies almost global stability and local asymptotical stability. The criterion has a remarkable application, which is used for asymptotical stability controller design. Further, combining with sum of squares (SOS) technique, this paper exploits a novel polynomial nonlinear controller design method. Examples illustrate the validity of the results.