Abstract
The relation between a class of high-order control variations and the asymptotic stabilizability of a smooth control system is briefly discussed. Assuming that there exist high-order control variations ”pointing” to a closed set at every point of some its neighborhood, an approach for constructing stabilizing feedback controls is proposed. Two illustrative examples are also presented.
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References
Agrachev, A., Gamkrelidze, R.: Local controllability and semigroups of diffeomorphisms. Acta Applicandae Mathematicae 32, 1–57 (1993)
Brockett, R.: Asymptotic stability and feedback stabilization. In: Brockett, R., Millmann, R., Sussmann, H. (eds.) Differential Geometric Control Theory. Progr. Math., vol. 27, pp. 181–191. Birkhäuser, Basel (1983)
Clarke, F.H., et al.: Nonsmooth analysis and control theory. Graduate Text in Mathematics, vol. 178. Springer, New York (1998)
Frankowska, H.: Local controllability of control systems with feedback. J. Optimiz. Theory Appl. 60, 277–296 (1989)
Hermes, H.: On the synthesis of a stabilizing feedback control via Lie algebraic methods. SIAM J. Control Optimiz. 16, 715–727 (1978)
Hermes, H.: Lie algebras of vector fields and local approximation of attainable sets. SIAM J. Control Optimiz. 18, 352–361 (1980)
Krastanov, M., Quincampoix, M.: Local small-time controllability and attainability of a set for nonlinear control systems. ESAIM: Control. Optim. Calc. Var. 6, 499–516 (2001)
Krastanov, M.I.: A sufficient condition for small-time local attainability of a set. Control and Cybernetics 31(3), 739–750 (2002)
Krastanov, M.I., Veliov, V.M.: On the controllability of switching linear systems. Automatica 41, 663–668 (2005)
Sussmann, H.J.: A general theorem on local controllability. SIAM J. Control Optimiz. 25, 158–194 (1987)
Veliov, V.M., Krastanov, M.I.: Controllability of piecewise linear systems. Systems & Control Letters 7, 335–341 (1986)
Veliov, V.: On the controllability of control constrained linear systems. Math. Balk., New Ser. 2, 147–155 (1988)
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Krastanov, M.I. (2007). Lie Brackets and Stabilizing Feedback Controls. In: Boyanov, T., Dimova, S., Georgiev, K., Nikolov, G. (eds) Numerical Methods and Applications. NMA 2006. Lecture Notes in Computer Science, vol 4310. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70942-8_39
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DOI: https://doi.org/10.1007/978-3-540-70942-8_39
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