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Stabilization of affine in control nonlinear systems. (English) Zbl 0662.93055

Asymptotic and practical stabilization is dealt with of single-input affine control systems by a feedback law. A main result states that a system \(\dot x=f(x)+u g(x)\) is asymptotically and practically stabilizable, if there exists a Lyapunov function V such that \(L_ gV=0\) implies \(L_ fV<0\) \((L_ f\) stands for the Lie derivative with respect to f).
It is not out of the place to mention here that the relevance of a condition like the above for deriving stabilizing feedbacks is well known in the literature, not only for single-input, but also for multi-input control systems [cf. E. B. Lee and L. Markus, Foundations of optimal control theory (1967; Zbl 0159.132) or V. M. Kuntsevich and M. M. Lychak, Synthesis of automatic control systems via Lyapunov functions (Russian) (1977; Zbl 0451.93001)].
Reviewer: K.Tchon

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
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
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[1] Sussmann, H. J., Sabanalytic sets and feedback control, J. diff. Eqns, 31, 31-52 (1979) · Zbl 0407.93010
[2] Sontag, E. D.; Sussmann, H. J., Remarks on continuous feedback, Proc. of CDC (1980)
[3] Jurdjevic, V.; Quinn, J. P., Controllability and stability, J. diff. Eqns, 28, 381-389 (1978) · Zbl 0417.93012
[4] Brockett, R. W., Asymptotic stability and feedback stabilization, (Brockett, R. W.; Millmann, R. S.; Sussmann, H. J., Differential Geometric Control Theory (1983), Birkhauser: Birkhauser Boston), 181-191 · Zbl 0528.93051
[5] Aeyels, D., Stabilization of a class of nonlinear systems by a smooth feedback control, Syst. Control Lett., 5, 289-294 (1985) · Zbl 0569.93056
[6] Aeyels, D., Stabilization by smooth feedback of the angular velocity of a rigid body, Syst. Control Lett., 5, 59-63 (1985) · Zbl 0566.93047
[7] Bacciotti, A., Poisson stabilizability via nonlinear feedback, Syst. Control Lett., 6, 390-394 (1982) · Zbl 0491.93048
[8] Bacciotti, A., Potentially global stabilizability, IEEE Trans., AC-31, 974-976 (1986) · Zbl 0605.93043
[9] Bacciotti, A., Remarks on the stabilizability problem for nonlinear systems, Proc. of CDC (1986) · Zbl 0605.93043
[10] Slemrod, M., Stabilization of bilinear control systems with applications to nonconservative problems in elasticity, SIAM J. Control Optim., 16, 132-141 (1978) · Zbl 0388.93037
[11] Kalouptsidis, N.; Tsinias, J., Stability improvement of nonlinear systems by feedback, IEEE Trans., AC-29, 364-367 (1984) · Zbl 0554.93057
[12] Tsinias, J., Stabilization of nonlinear control systems to subspaces, Int. J. Control, 46, 529-538 (1987) · Zbl 0629.93049
[13] Erratum, Ann. Math., 68, 202 (1958) · Zbl 0081.08601
[14] Brickell, F.; Clark, R. S., Differentiable Manifolds (1970), Van Nontrand: Van Nontrand New York · Zbl 0199.56303
[15] Wonham, W. M., Linear Multivariable Control: A Geometric Approach (1985), Springer: Springer New York · Zbl 0393.93024
[16] Aeyels̈, D., Local and global controllability for nonlinear systems, Syst. Control Lett., 5, 19-62 (1984) · Zbl 0552.93009
[17] Kalouptsidis, N.; Elliot, D., Stability analysis of the orbits of control systems, Math. Syst. Theory, 15, 323-342 (1982) · Zbl 0464.93036
[18] Bacciotti, A.; Kalouptsidis, N., Topological dynamics of control systems: stability and attraction, Nonlinear Analysis, 10, 547-565 (1986) · Zbl 0611.93052
[19] Tsinias, J.; Kalouptsidis, N.; Bacciotti, A., Lyapunov functions and stability of dynamical polysystems, Math. Syst. Theory, 19, 333-354 (1987) · Zbl 0628.93056
[20] Tsinias, J.; Kalouptsidis, N., Prolongations and stability analysis via Lyapunov functions of dynamical polysystems, Math. Syst. Theory, 20, 215-233 (1987) · Zbl 0642.93052
[21] TsiniasNonlinear Analysis; TsiniasNonlinear Analysis
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