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On the existence of stabilising feedback controls for real analytic small-time locally controllable systems

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Abstract

It is shown that, for real analytic control systems of the form \(f:{M}\times \varOmega \ni (q,u)\mapsto f(q,u)\in {T}_q{M}\), where M is a real analytic manifold and \(\varOmega \) is a separable metric space, small-time local controllability from an equilibrium \(p\in {M}\) implies the existence of a piecewise analytic feedback control that locally stabilises f at p. The proof is similar in spirit to an earlier analogous result for globally controllable systems; however, it resolves several technical obstructions that emerge when the assumption of small-time local controllability is substituted for that of global controllability. In the light of a recent characterisation of small-time local controllability for homogeneous control systems, the main result of the paper implies that, for a large class of control systems that appear in applications and the literature, there is a computable sufficient condition for stabilisability by means of a piecewise analytic feedback control.

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Notes

  1. In the technical sense of the word. That is, patchy feedback controls that yield a cost within \(\varepsilon >0\) of the optimum, for any \(\varepsilon >0\).

  2. That is, roughly speaking, when convergence in (3) occurs exponentially fast.

  3. Since \(\varOmega \) is only a metric space, a clarification is in order here: the notation \({T}f_{\omega }\) means that we fix \(\omega \) to compute the tangent map of \(f_{\omega }\) and subsequently consider the resulting map on \(\hbox {TM}\times \varOmega \).

  4. The elements of a semi-group need not have inverses and this is the case here because the parameters \(t_i\) are non-negative.

  5. A point x of a subanalytic subset N of a manifold M is called smooth of dimension k if there exists a neighbourhood U of x in M such that \({N}\cap {{U}}\) is an analytic submanifold of M of dimension k. The dimension of N is defined to be the maximum of the dimensions of the smooth points of N.

  6. Small-time local controllability is the main premise throughout the paper, hence the interest in this implication.

  7. We exclude the point p because, at p, the partition is not locally finite: every neighbourhood of p contains infinitely many sets \({H}_j\). For every point \(q\ne p\), on the other hand, there exists a sufficiently small neighbourhood around q that intersects finitely many sets \({H}_j\).

  8. The size of such sets is irrelevant since the problem we are dealing with is of an infinitesimal nature. In other words, it suffices to be able to construct the feedback in some neighbourhood of the equilibrium.

  9. The assumption of STLC for f requires that \(\varOmega \) satisfy additional properties [46, 48].

  10. To make any comparison between asymptotic controllability and small-time local controllability in a meaningful way, we have to restrict the former to a neighbourhood of an equilibrium since small-time local controllability is a local, in fact infinitesimal, property. However, this modification of the definition of asymptotic controllability is straightforward.

  11. That is, invariant under a one-parameter family of dilations.

  12. See Problem 10.4 in [14, p. 315] and the references therein for the long history of vector-valued quadratic forms and their relevance to control theory.

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Acknowledgments

The author would like to thank Dr M. I. Krastanov for useful and detailed comments on the present work.

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  1. Faculty of Aerospace Engineering, The Technion—Israel Institute of Technology, Haifa, Israel

    Pantelis Isaiah

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Isaiah, P. On the existence of stabilising feedback controls for real analytic small-time locally controllable systems. Math. Control Signals Syst. 27, 467–492 (2015). https://doi.org/10.1007/s00498-015-0148-z

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