##
**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/observation systems with incomplete information |

### Keywords:

Global stabilization; Finite-time convergence; Hölder continuous state feedback; Lyapunov stability; Uncertain nonlinear systems
Full Text:
DOI

### References:

[1] | Athans, M.; Falb, P. L., Optimal control: An introduction to theory and its applications (1966), McGraw-Hill: McGraw-Hill New York · Zbl 0186.22501 |

[2] | Bacciotti, A., Local stabilizability of nonlinear control systems (1992), World Scientific: World Scientific Singapore · 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).; 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).; 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). |

[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), Birkhauser: Birkhauser Boston · Zbl 0857.93001 |

[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, (Elaydi, S., Differential equations stability and control (1991), Marcel Dekker: Marcel Dekker New York), 249-260 · Zbl 0711.93069 |

[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) · Zbl 0974.93050 |

[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 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.