### Sufficient Conditions for Asymptotic Controllability and Hybrid Feedback Stabilization for a Class of Time-Varying Systems

J. Tsinias
2007 IEEE Transactions on Automatic Control
For a class of time-varying systems, including those whose dynamics have triangular structure, we provide sufficient conditions for global asymptotic controllability and global asymptotic stabilization by means of a hybrid time-varying feedback. Index Terms-Global asymptotic controllability, hybrid feedback stabilization, persistence of excitation, triangular systems. Notations: For given x 2 n ; x 0 denotes its transpose and jxj its Euclidean norm. We denote by jAj := maxfjAxj : x 2 n ; jxj =
more » ... xj : x 2 n ; jxj = 1g the induced norm of a matrix A 2 m2n . By N we denote the set of all C o functions ' : + := [0; 1) ! + which are nondecreasing with '(s) > 0 for all s > 0 and by K the set of all strictly increasing ' 2 N satisfying '(0) = 0. By L we denote the set of all decreasing C o functions ' : + ! + with '(s) ! 0 as s ! 1. We say that a function R : + 2 + 2 + ! + is of class LN N , if for each fixed (t; s) 2 + 2 + , the functions R(1; t; s); R(t; 1; s); R(t; s; 1) are of class L; N; N , respectively, and we say that is of class NNN , when for each fixed (t; s) 2 + 2 + , the functions R(1; t; s); R(t; 1; s); R(t; s; 1) are of class N . Finally, for each pair of subsets I 1 ; I 2 n , we adopt the notation L(I 1 ; I 2 ) to denote the class of all measurable and locally bounded functions ' : I1 ! I2.