Analysis of a singular point of control system with piecewise smooth feedback law

Assigning instability to some singular points is useful for global asymptotic stabilization if a nonlinear control system is not globally asymptotically stabilizable to a single singular point. It is shown that the singular points are expressed by the intersection of submanifolds of X/spl times/U. If the linear approximation is controllable, the Conley index of the singular point can be homotopy equivalent to that of a hyperbolic singular point with any dimensional unstable manifold by using the discontinuous feedback.