Global stabilization of discrete-time homogeneous systems

The feedback stabilization of continuous time Delta (gamma)-homogeneous sys tems has been studied by several authors, see for instance (Dayawansa and M artin, proceedings of the 28th C.D.C., Tampa, Florida, December, 1989, Herm es, J. Differential Equations 92 (1991) 76-89, Hermes, Systems Control Lett . 24 (1995) 7 -11, Kawski, Proceedings of the 27th C.D.C., Texas, 1988, Tex as, Kawski, Control Theory and Advanced Technology, vol. 6, MITA Press, Tok yo, 1996, pp. 497-516, Sepulchre and Aeyels, Math. Control Signal Systems 9 (1996) 34-58). In (Kawski, Analysis of Controlled Dynamical Systems, Birkh auser, Basel, 1991, Kawski, Proceedings of IFAC NOLCOS, 1995), Kawski gave an intrinsic concept of homogeneity with respect to general smooth dilation s. In this note, we define the homogeneity of transition maps w.r.t. contin uous dilations and we show how one can construct a globalstabilizing feedba ck control for discrete-time homogeneous systems which are locally asymptot ically stabilizable. (C) 1999 Elsevier Science B.V. All rights reserved.