Finite time stability and stabilization of a class of continuous systems. (English) Zbl 1131.93043

The paper deals with systems of ODE’s in finite-dimensional space having unique solutions in forward time. It discusses finite-time stability, i.e., the strong version of asymptotic stability when the systems reaches the equilibrium point. The contains two main results. The first result shows that, under appropriate assumptions, existence of a Lyapunov function plus a certain integral property are necessary and sufficient for finite time stability of a system of ODE’s. The second result shows that a control-affine system admits a feedback making it finite time stable if and only if there exists a control Lyapunov function satisfying a certain differential inequality. In that case, the finite time stabilizing feedback is given in explicit form.


93D15 Stabilization of systems by feedback
93D30 Lyapunov and storage functions
93C10 Nonlinear systems in control theory
93C15 Control/observation systems governed by ordinary differential equations
Full Text: DOI


[1] Lyapunov, A.M., Stability of motion: general problem, Internat. J. control, 55, 3, 520-790, (1992), (Lyapunov centenary issue)
[2] Floquet, T.; Barbot, J.P.; Perruquetti, W., Higher-order sliding mode stabilization for a class of nonholonomic perturbed systems, Automatica J. IFAC, 39, 6, 1077-1083, (2003) · Zbl 1038.93063
[3] Perruquetti, W.; Barbot, J.P., Sliding mode control in engineering, (2002), Dekker
[4] Haimo, V.T., Finite time controllers, SIAM J. control optim., 24, 4, 760-770, (1986) · Zbl 0603.93005
[5] Bhat, S.P.; Bernstein, D.S., Finite time stability of continuous autonomous systems, SIAM J. control optim., 38, 3, 751-766, (2000) · Zbl 0945.34039
[6] Hong, Y., Finite-time stabilization and stabilizability of a class of controllable systems, Systems control lett., 46, 231-236, (2002) · Zbl 0994.93049
[7] Hong, Y.; Xu, Y.; Huang, J., Finite-time control for robot manipulators, Systems control lett., 46, 243-253, (2002) · Zbl 0994.93041
[8] W. Perruquetti, S. Drakunov, Finite time stability and stabilisation, in: IEEE Conference on Decision and Control, Sydney, Australia, 2000
[9] Sontag, E., A universal construction of Arststein’s theorem on nonlinear stabilization, Systems control lett., 13, 117-123, (1989) · Zbl 0684.93063
[10] Agarwal, R.P.; Lakshmikantham, V., Uniqueness and nonuniqueness criteria for ordinary differential equations, Ser. real anal., vol. 6, (1993), World Scientific Singapore · Zbl 0785.34003
[11] Filippov, A.F., Differential equations with discontinuous right-hand sides, (1988), Kluwer Academic Dordrecht · Zbl 0664.34001
[12] Kawski, M., Stabilization of nonlinear systems in the plane, Systems control lett., 12, 169-175, (1989) · Zbl 0666.93103
[13] Kurzweil, J., On the inversion of Lyapunov’s second theorem on stability of motion, Amer. math. soc. transl., 24, 19-77, (1963) · Zbl 0127.30703
[14] Hahn, W., Theory and application of Lyapunov’s direct method, (1963), Prentice Hall · Zbl 0119.07403
[15] Clarke, F.H.; Ledyaev, Y.S.; Stern, R.J., Asymptotic stability and smooth Lyapunov function, J. differential equations, 149, 69-114, (1998) · Zbl 0907.34013
[16] Teel, A.R.; Praly, L., A smooth Lyapunov function from a class-KL estimate involving two positive semidefinite functions, ESAIM control optim. calc. var., 5, 313-367, (2000) · Zbl 0953.34042
[17] E. Moulay, W. Perruquetti, Finite time stability of nonlinear systems, in: IEEE Conference on Decision and Control, Hawaii, USA, 2003, pp. 3641-3646
[18] Artstein, Z., Stabilization with relaxed controls, Nonlinear anal., 7, 11, 1163-1173, (1983) · Zbl 0525.93053
[19] Mickael, E., Continuous selections, Anal. math., 63, 2, 361-382, (1956) · Zbl 0071.15902
[20] Aubin, J.P.; Frankowska, H., Set-valued analysis, (1990), Springer-Verlag New York
[21] S.P. Bhat, D. Bernstein, Continuous, bounded, finite-time stabilization of the translational and rotational double integrator, in: IEEE Conference on Control Applications, Dearborn, MI, 1996, pp. 185-190
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.