Abstract
The leading term of a control variation is given, in general, by a formula that involves complicated iterated integrals and does not explicitly exhibit the fact that the result is a tangent vector. Agrachev, Gamkrelidze and Sarychev have proposed a formalism, based on their “chronological calculus,” that makes it possible to show that the leading term is always equal to a Lie bracket plus other expressions that vanish at the given reference point. In this note we propose an alternative approach, based on classical facts of Lie theory, such as the Campbell-Hausdorff Formula and a graded version B of the classical projection map from associative polynomials to Lie polynomials. We refer to B as the bracketization map. We derive a general formula that says that for any exponential Lie series on a graded linear space, each homogeneous component is equal to its own bracketization plus a noncommutative polynomial in the bracketizations of the preceding components. This implies that the first term of the series that does not vanish at a point is equal, at that point, to its own bracketization.
Keywords
- Formal Power Series
- Partial Differential Operator
- Homogeneous Element
- Homogeneous Component
- Leading Term
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This work was supported in part by the National Science Foundation under NSF Grant DMS-8902994.
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References
Agrachev, A.A., and R.V. Gamkrelidze, “The exponential representation of flows and chronological calculus,” Math. Sbornik 107 (149) (1978), pp. 467–532; Engl. Transi, in Math. USSR Sbornik 35 (1979), pp. 727–785.
Agrachev, A.A., R.V. Gamkrelidze and A.V. Sarychev, “Local invariant of smooth control systems,” Acta Applicandae Mathematicae 14 (1989), pp. 191–237.
Bourbaki, N., Groupes et Algèbres de Lie, Élements de Mathématique, Fascicule XXXVII, Chap. II et III, Hermann, Paris, 1972.
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© 1991 Birkhäuser Boston
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Sussmann, H.J. (1991). The Leading Term of a Control Variation. In: Bonnard, B., Bride, B., Gauthier, JP., Kupka, I. (eds) Analysis of Controlled Dynamical Systems. Progress in Systems and Control Theory, vol 8. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-3214-8_34
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DOI: https://doi.org/10.1007/978-1-4612-3214-8_34
Publisher Name: Birkhäuser Boston
Print ISBN: 978-1-4612-7835-1
Online ISBN: 978-1-4612-3214-8
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