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A continuous feedback approach to global strong stabilization of nonlinear systems. (English) Zbl 1012.93053

The paper addresses the problem of the global stabilization by means of continuous feedback of a class of nonlinear (possibly non-affine and nonsmooth) systems which in general cannot be stabilized by smooth (i.e., at least C 1 ) feedback. The basic result concerns systems which can be represented as a chain of power integrators perturbed by a vector field in triangular form

x ˙ 1 =d 1 (t)x 2 p 1 +f 1 (t,x 1 ,x 2 )x ˙ i =d i (t)x i+1 p i +f i (t,x 1 ,,x i+1 )x ˙ n =d n (t)u p n +f n (t,x 1 ,,x n ,u)

where the p i ’s are odd, the d i (t) are unknown but constrained to a bounded interval, and

f i (t,x 1 ,,x i+1 )= j=0 p i -1 x i+1 j a ij (t,x 1 ,,x i )

(x n+1 stands for u). The functions a ij are subject to some other technical assumptions. The proof is based on an iterative procedure and exploits the theory of homogeneous systems. It uses the method of adding a power integrator in order to explicitly construct a continuous feedback and generate a C 1 Lyapunov function. The paper contains also some extensions of the basic result and a rich variety of interesting examples.

MSC:
93D15Stabilization of systems by feedback
93D30Scalar and vector Lyapunov functions
93C10Nonlinear control systems