Global stabilization of homogeneous polynomial systems in $R^n$
Résumé
This paper deals with a nonlinear control process dx/dt=X(x)+Bu, x∈\bold Rⁿ, u∈ \bold R^n-1, where X(x) is a homogeneous polynomial function. The authors present a necessary and sufficient condition for this system to be globally stabilizable by an analytic in x feedback u(x) (u(0)=0). The stabilizing feedback is constructed explicitly for even and odd functions X(x). Illustrative examples are given. For a control system of the form \dot x=X(x)+Bu, x∈\bbfR\sp n, u∈\bbfR\spn-1, where X is a homogeneous vector field and B a matrix of rank n-1, the problem of global stabilizability by feedback control u=u(x) is considered. A necessary and sufficient condition for such a control is obtained, and an explicit construction of it is given. For polynomials of odd degree this construction is satisfactory, and can also be used for local stabilizability of non-homogeneous polynomials. For even degree, the construction does not work, but the authors promise another future paper for this case.