Smooth global stabilization for a class of nonlinear systems using homogeneity with monotone degrees

This paper considers the problem of designing C¹ (continuously differentiable) state feedback stabilizers for a class of 3-dimensional nonlinear systems whose linearizations around the origin may contain uncontrollable modes. Based on a new definition of homogeneity with monotone degrees, we not only propose conditions of constructing C¹ and C^∞ (smooth) controllers, but also provide explicit design schemes for such systems. Several examples are investigated to show the advantages of the generalized homogeneity compared to the traditional unified homogeneity.