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Non-Lipschitz continuous stabilizers for nonlinear systems with uncontrollable unstable linearization. (English) Zbl 0974.93050
Summary: Without imposing any growth condition, we prove that every chain of odd power integrators perturbed by a $$C^{1}$$ triangular vector field is globally stabilizable via non-Lipschitz continuous state feedback, although it is not stabilizable, even locally, by any smooth state feedback because the Jacobian linearization may have uncontrollable modes whose eigenvalues are on the right half-plane. The proof is constructive and accomplished by developing a machinery – adding a power integrator – that enables one to explicitly design a $$C^{0}$$ globally stabilizing feedback law as well as a $$C^{1}$$ control Lyapunov function which is positive definite and proper.

##### MSC:
 93D15 Stabilization of systems by feedback 93C10 Nonlinear systems in control theory
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##### References:
 [1] Bacciotti, A., Local stabilizability of nonlinear control systems, (1992), World Scientific Singapore · Zbl 0757.93061 [2] Brockett, R.W., Asymptotic stability and feedback stabilization, (), 181-191 · Zbl 0528.93051 [3] Byrnes, C.I.; Isidori, A., New results and examples in nonlinear feedback stabilization, Systems control lett., 12, 437-442, (1989) · Zbl 0684.93059 [4] Celikovsky, S.; Aranda-Bricaire, E., Constructive non-smooth stabilization of triangular systems, Systems control lett., 36, 21-37, (1999) · Zbl 0913.93057 [5] Coron, J.M.; Praly, L., Adding an integrator for the stabilization problem, Systems control lett., 17, 89-104, (1991) · Zbl 0747.93072 [6] W.P. Dayawansa, Recent advances in the stabilization problem for low dimensional systems, Proceedings of second IFAC Symposium on Nonlinear Control Systems Design Symposium, Bordeaux, 1992, pp. 1-8. [7] Dayawansa, W.P.; Martin, C.F.; Knowles, G., Asymptotic stabilization of a class of smooth two dimensional systems, SIAM J. control optim., 28, 1321-1349, (1990) · Zbl 0731.93076 [8] Hahn, W., Stability of motion, (1967), Springer Berlin · Zbl 0189.38503 [9] Hermes, H., Homogeneous coordinates and continuous asymptotically stabilizing feedback controls, (), 249-260 [10] Hermes, H., Nilpotent and high-order approximations of vector field systems, SIAM rev., 33, 238-264, (1991) · Zbl 0733.93062 [11] Isidori, A., Nonlinear control systems, (1995), Springer New York · Zbl 0569.93034 [12] Kawski, M., Stabilization of nonlinear systems in the plane, Systems control lett., 12, 169-175, (1989) · Zbl 0666.93103 [13] Kawski, M., Homogeneous stabilizing feedback laws, Control theory adv. technol., 6, 497-516, (1990) [14] Qian, C.; Lin, W., A continuous feedback approach to global strong stabilization of nonlinear systems, IEEE trans. automat. control, 46, 6, (2001) [15] Kurzweil, J., On the inversion of Lyapunov’s second theorem on the stability of motion, Amer. math. soc. transl. ser. 2, 24, 19-77, (1956) [16] Lin, Y.; Sontag, E.; Wang, Y., A smooth Lyapunov theorem for robust stability, SIAM J. control optim., 28, 1491-1503, (1996) [17] Lin, W.; Qian, C., Adding one power integrator: a tool for global stabilization of high-order lower-triangular systems, Systems control lett., 39, 339-351, (2000) · Zbl 0948.93056 [18] Rosier, L., Homogeneous Lyapunov function for homogeneous continuous vector field, Systems control lett., 19, 467-473, (1992) · Zbl 0762.34032 [19] C. Rui, M. Reyhanoglu, I. Kolmanovsky, S. Cho, N.H. McClamroch, Non-smooth stabilization of an underactuated unstable two degrees of freedom mechanical system, Proceedings of the 36th IEEE CDC, San Diego, 1997, pp. 3998-4003. [20] Sontag, E.D., Feedback stabilization of nonlinear systems, (), 61-81 · Zbl 0735.93063 [21] Sontag, E.D., A “universal” construction of Artstein’s theorem on nonlinear stabilization, Systems control lett., 13, 117-123, (1989) · Zbl 0684.93063 [22] G. Stefani, Polynomial approximations to control systems and local controllability, Proceedings of the 24th IEEE CDC, Florida, 1985, pp. 33-38. [23] Sussmann, H., A general theorem on local controllability, SIAM J. control optim., 25, 158-194, (1987) · Zbl 0629.93012 [24] Tsinias, J., Sufficient Lyapunov-like conditions for stabilization, Math. control signals systems, 2, 343-357, (1989) · Zbl 0688.93048 [25] Tzamtzi, M.; Tsinias, J., Explicit formulas of feedback stabilizers for a class of triangular systems with uncontrollable linearization, Systems control lett., 38, 115-126, (1999) · Zbl 1043.93548
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