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Local controllability of control systems with feedback

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Abstract

We prove sufficient conditions for the instantaneous local controllability of nonlinear (nonsmooth) control systems with feedback. We introduce for that purpose the high-order variations of the reachable map, which is related strongly to its shape. It is a direction in which the reachable map evolves. The cone of variations is convex. This allows one to prove the following theorem: if there exist variationsv 1, ...,v p such that ϑ ∈ int co{v 1, ...,v p }, then the system is small-time locally controllable at the point of equilibrium. We provide a short proof of the main result of Sussmann (Ref. 12) and extend it to differential inclusions.

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References

  1. Frankowska, H.,An Open Mapping Principle for Set-Valued Maps, Journal of Mathematical Analysis and Applications, Vol. 127, pp. 172–180, 1987.

    Google Scholar 

  2. Hermes, H.,Control Systems Which Generate Decomposable Lie Algebras, Journal of Differential Equations, Vol. 44, pp. 166–187, 1982.

    Google Scholar 

  3. Bianchini, R. M., andStefani, G.,Local Controllability and Bilinearization. IMA Journal of Applied Mathematics, Vol 1, pp. 173–183, 1984.

    Google Scholar 

  4. Filippov, A. F.,Classical Solutions of Differential Equations with Multivalued Right-Hand Side, SIAM Journal on Control and Optimization, Vol. 5, pp. 609–621, 1967.

    Google Scholar 

  5. Clarke, F. H.,Nonsmooth Analysis and Optimization, Wiley Interscience, New York, New York, 1983.

    Google Scholar 

  6. Aubin, J. P., andCellina, A.,Differential Inclusions, Springer-Verlag, Berlin, Germany, 1984.

    Google Scholar 

  7. Haynes, G. W., andHermes, H.,Nonlinear Controllability via Lie Theory, SIAM Journal on Control and Optimization, Vol. 8, pp. 450–460, 1970.

    Google Scholar 

  8. Lobry, C.,Controlabilité des Systèmes Non Linéaires, SIAM Journal on Control and Optimization, Vol. 8, pp. 573–605, 1970.

    Google Scholar 

  9. Brockett, R. W.,System Theory on Group Manifolds and Coset Spaces, SIAM Journal on Control and Optimization, Vol. 10, pp. 265–284, 1972.

    Google Scholar 

  10. Krener, A.,A Generalization of Chow's Theorem and the Bang-Bang Theorem to Nonlinear Control Problems, SIAM Journal on Control and Optimization, Vol. 12, pp. 43–52, 1974.

    Google Scholar 

  11. Hermann, R., andKrener, A. J.,Nonlinear Controllability and Observability, IEEE Transactions on Automatic Control, Vol. AC-22, pp. 728–740, 1977.

    Google Scholar 

  12. Sussmann, H.,A Sufficient Condition for Local Controllability, SIAM Journal on Control and Optimization, Vol. 16, pp. 790–802, 1978.

    Google Scholar 

  13. Sussmann, H.,Lie Brackets and Local Controllability: a Sufficient Condition for Scalar-Input Systems, SIAM Journal on Control and Optimization, Vol. 21, pp. 686–713, 1983.

    Google Scholar 

  14. Sussmann, H.,Lie Brackets, Real Analyticity, and Geometric Control. Differential Geometric Control Theory, Edited by R. W. Brockett, R. S. Milman, and H. J. Sussmann, Birkhauser, Boston, Massachusetts, pp. 1–116, 1983.

    Google Scholar 

  15. Sussmann, H., andJurdevic, V.,Controllability of Nonlinear Systems, Journal of Differential Equations, Vol. 12, pp. 95–116, 1972.

    Google Scholar 

  16. Sussmann, H.,A General Theorem on Local Controllability, SIAM Journal on Control and Optimization, Vol. 25, pp. 158–194, 1987.

    Google Scholar 

  17. Aubin, J. P., andEkeland, I.,Applied Nonlinear Analysis, Wiley Interscience, New York, New York, 1984.

    Google Scholar 

  18. Aubin, J. P., andFrankowska, H.,Heavy Viable Trajectories of Controlled Systems, Annales de l'Institut Henri Poincaré, Analyse Non Linéaire, Paris, France, Vol. 2, pp. 371–395, 1985.

    Google Scholar 

  19. Frankowska, H.,The Maximum Principle for an Optimal Solution to a Differential Inclusion with End Point Constraints, SIAM Journal on Control and Optimization, Vol. 25, pp. 145–157, 1987.

    Google Scholar 

  20. Frankowska, H.,Local Controllability and Infinitesimal Generators of Semigroups of Set-Valued Maps. SIAM Journal on Control and Optimization, Vol. 25, pp. 412–432, 1987.

    Google Scholar 

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Authors and Affiliations

  1. CNRS, National Scientific Research Council of France, Paris, France

    H. Frankowska (Chargé de Recherche)

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  1. H. Frankowska
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Communicated by R. Conti

This work was supported in part by FCAR of Canada.

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Frankowska, H. Local controllability of control systems with feedback. J Optim Theory Appl 60, 277–296 (1989). https://doi.org/10.1007/BF00940008

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