# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [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, (), 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] Massera, J.L., Erratum, Ann. math., Ann. math., 68, 202-206, (1958) · Zbl 0081.08601 [14] Brickell, F.; Clark, R.S., Differentiable manifolds, (1970), Van Nontrand New York · Zbl 0199.56303 [15] Wonham, W.M., Linear multivariable control: A geometric approach, (1985), 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] T\scsinias J., A Lyapunov description of stability in control systems, Nonlinear Analysis (to appear).
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.