Asymptotic stability and feedback stabilization

We consider the loca1 behavior of control problems described by (* = dx/dr) i=f(x,u) ; f(x,0)=0 o and more specifically, the question of determining when there exists a smooth function u(x) such that x = xo is an equilibrium point which is asymPtotically stable. Our main results are formulated in Theorems 1 and 2 be1ow. Whereas it night have been suspected that controllability would insure the exlstence of a stabilizing control law, Theorem I uses a degree-theoretic argument to show this is far from being the case. The positive result of Theorem 2 can be thought of as providing an application of high gain feedback in a nonlinear setting. 1. Introduction In this paper we establish general theorems which are strong enough to irnply, among other things, that a) there is a continuous control law (u,v) = (u(xry, z) rv(x,y,z)) which makes the origin asympEoticatly stable for x=u y=v z=xy and that b) there exists no continuous control law (urv) = (u(xryrz), v(x,y,z)) which makes the origin asymptotically stable for 181