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Global finite-time stabilization of a class of uncertain nonlinear systems. (English) Zbl 1098.93032
Summary: This paper studies the problem of finite-time stabilization for nonlinear systems. We prove that global finite-time stabilizability of uncertain nonlinear systems that are dominated by a lower-triangular system can be achieved by Hölder continuous state feedback. The proof is based on the finite-time Lyapunov stability theorem and the nonsmooth feedback design method developed recently for the control of inherently nonlinear systems that cannot be dealt with by any smooth feedback. A recursive design algorithm is developed for the construction of a Hölder continuous, global finite-time stabilizer as well as a $C^1$ positive definite and proper Lyapunov function that guarantees finite-time stability.

##### MSC:
 93D15 Stabilization of systems by feedback 93C41 Control problems with incomplete information
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##### References:
 [1] Athans, M.; Falb, P. L.: Optimal control: an introduction to theory and its applications. (1966) · Zbl 0196.46303 [2] Bacciotti, A.: Local stabilizability of nonlinear control systems. (1992) · Zbl 0757.93061 [3] Bhat, S. P., & Bernstein, D. S. (1997). Finite-time stability of homogeneous systems. In Proceedings of the American control conference (pp. 2513-2514). [4] Bhat, S. P.; Bernstein, D. S.: Continuous finite-time stabilization of the translational and rotational double integrators. IEEE transactions on automatic control 43, 678-682 (1998) · Zbl 0925.93821 [5] Bhat, S. P.; Bernstein, D. S.: Finite-time stability of continuous autonomous systems. SIAM journal on control and optimization 38, 751-766 (2000) · Zbl 0945.34039 [6] Coron, J. M.; Praly, L.: Adding an integrator for the stabilization problem. System and control letters 17, 89-104 (1991) · Zbl 0747.93072 [7] Dayawansa, W. (1992). Recent advances in the stabilization problem for low dimensional systems. In Proceedings of the second IFAC symposium on nonlinear control systems and design, Bordeaux (pp. 1-8). · Zbl 0976.93509 [8] Dayawansa, W.; Martin, C.; Knowles, G.: Asymptotic stabilization of a class of smooth two dimensional systems. SIAM journal on control and optimization 28, 1321-1349 (1990) · Zbl 0731.93076 [9] Freeman, R.; Kokotovic, P.: Robust nonlinear control design: state-space and Lyapunov techniques. (1996) · Zbl 0863.93075 [10] Haimo, V. T.: Finite-time controllers. SIAM journal on control and optimization 24, 760-770 (1986) · Zbl 0603.93005 [11] Hermes, H.: Homogeneous coordinates and continuous asymptotically stabilizing feedback controls. Differential equations stability and control, 249-260 (1991) [12] Hong, Y.: Finite-time stabilization and stabilizability of a class of controllable systems. System & control letters 46, 231-236 (2002) · Zbl 0994.93049 [13] Hong, Y.; Huang, J.; Xu, Y.: On an output feedback finite-time stabilization problem. IEEE transactions on automatic control 46, 305-309 (2001) · Zbl 0992.93075 [14] Kawski, M.: Stabilization of nonlinear systems in the plane. System & control letters 12, 169-175 (1989) · Zbl 0666.93103 [15] Kawski, M.: Homogeneous stabilizing feedback laws. Control theory and advanced technology 6, 497-516 (1990) [16] Praly, L.; Andrea-Novel, B.; Coron, J.: Lyapunov design of stabilizing controllers for cascaded systems. IEEE transactions on automatic control 36, 1177-1181 (1991) [17] Qian, C.; Lin, W.: A continuous feedback approach to global strong stabilization of nonlinear systems. IEEE transactions on automatic control 46, 1061-1079 (2001) · Zbl 1012.93053 [18] Qian, C.; Lin, W.: Non-Lipschitz continuous stabilizer for nonlinear systems with uncontrollable unstable linearization. System & control letters 42, 185-200 (2001) [19] Rosier, L.: Homogeneous Lyapunov function for homogeneous continuous vector field. System & control letters 19, 467-473 (1992) · Zbl 0762.34032 [20] Ryan, E. P.: Singular optimal controls for second-order saturating system. International journal of control 36, 549-564 (1979) · Zbl 0422.49006