×

zbMATH — the first resource for mathematics

A necessary condition for feedback stabilization. (English) Zbl 0699.93075
Summary: We give a new necessary condition for the existence of a - static or dynamic - continuous asymptotically stabilizing feedback control for the system \(\dot x=f(x,u)\). We apply our condition to give an example of a system in \({\mathbb{R}}^ 3\) which is small time locally controllable, satisfies ‘Brockett’s second and third condition’ [see R. W. Brockett, Differential geometric control theory, Proc. Conf., Mich. Technol. Univ. 1982, Prog. Math. 27, 181-191 (1983; Zbl 0528.93051)] and cannot be locally asymptotically stabilized by employing a - static or dynamic - continuous feedback law.

MSC:
93D15 Stabilization of systems by feedback
93C10 Nonlinear systems in control theory
93C15 Control/observation systems governed by ordinary differential equations
93D20 Asymptotic stability in control theory
Citations:
Zbl 0528.93051
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aeyels, D., Stabilization of a class of nonlinear systems by a smooth feedback control, Systems control lett., 5, 289-294, (1985) · Zbl 0569.93056
[2] L. Praly, B. d’Andréa-Novel and J.M. Coron, Lyapunov design of stabilizing controllers for cascaded systems, Preprint.
[3] Artstein, Z., Stabilization with relaxed controls, Nonlinear anal. theory meth. appl., 7, 1163-1173, (1983) · Zbl 0525.93053
[4] Brockett, R.W., Asymptotic stability and feedback stabilization, () · Zbl 0528.93051
[5] Brunovsky, P., Local controllability of odd systems, Banach center publications (Warsaw), 1, 39-45, (1974)
[6] Dayawansa, W.P.; Martin, C.F., Asymptotic stabilization of two dimensional real analytic systems, Systems control lett., 12, 205-211, (1989) · Zbl 0673.93064
[7] Hermes, H., Controlled stability, Annali mat. pura appl., 114, 103-119, (1977) · Zbl 0385.93010
[8] H. Hermes, Asymptotically stabilizing feedback controls and the nonlinear regulator problem, Preprint. · Zbl 0738.93061
[9] Hermes, H.; Kawski, M., Local controllability of a single-input affine system, (), 235-248
[10] Kawski, M., Control variations with an increasing number of switchings, Bull. amer. math. soc., 18, 149-152, (1988) · Zbl 0663.93006
[11] Kawski, M., Stabilization of nonlinear systems in the plane, Systems control lett., 12, 169-175, (1989) · Zbl 0666.93103
[12] M. Kawski, Homogeneous stabilizing feedback laws, Preprint. · Zbl 0736.93020
[13] Krasnoselskii, M.A.; Zabreiko, P.P., Geometric methods in nonlinear analysis, (1983), Springer-Verlag Berlin
[14] Krener, A., A generalization of Chow’s theorem and the bang-bang theorem to nonlinear control systems, SIAM J. control optim., 12, 43-52, (1974) · Zbl 0243.93008
[15] Sontag, E.D., A ‘universal’ construction of Artstein’s theorem on nonlinear stabilization, Systems control lett., 13, 117-123, (1989) · Zbl 0684.93063
[16] Sontag, E.D., Feedback stabilization of nonlinear systems, () · Zbl 0758.93013
[17] Sontag, E.D.; Sussmann, H.J., Remarks on continuous feedback, Ieee cdc, Vol. 2, 916-921, (1980), Albuquerque
[18] Spanier, E.H., ()
[19] Stefani, G., Polynomial approximations to control systems and local controllability, (), 33-38
[20] Sussmann, H.J., Subanalytic sets and feedback control, J. differential equations, 31, 31-52, (1979) · Zbl 0407.93010
[21] Sussmann, H.J., Lie brackets and local controllability: A sufficient conditions for scalar input systems, SIAM J. control optim., 21, 686-713, (1983) · Zbl 0523.49026
[22] Sussmann, H.J., A general theorem on local controllability, SIAM J. control optim., 25, 158-194, (1987) · Zbl 0629.93012
[23] Sussmann, H.J.; Jurdjevic, V., Controllability of nonlinear systems, J. differential equations, 12, 95-116, (1972) · Zbl 0242.49040
[24] Zabczyk, J., Some comments on stabilizability, Appl. math. optim, 19, 1-9, (1989) · Zbl 0654.93054
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.