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Sur la structure de l’ensemble d’accessibilité de certains systèmes: Application à l’equivalence des systèmes. (On the structure of the accessibility set of certain systems: Application to the equivalence of systems). (French) Zbl 0574.49023

The topological properties of accessibility sets for a large class of nonlinear systems are investigated. In particular, it is shown that accessible sets of self-invariant systems have a simple topological structure (the interior and the interior of the closure coincide). As a consequence, systems which are equivalent (in an appropriate sense) have accessible sets with the same interior and the same boundary. This is used to prove a conjecture by L. R. Hunt.
Reviewer: A.Bacciotti

MSC:

93B03 Attainable sets, reachability
93B05 Controllability
93C10 Nonlinear systems in control theory
57R27 Controllability of vector fields on \(C^\infty\) and real-analytic manifolds
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[1] R. W. Brockett, Non linear systems and differential geometry,Proc. IEEE, 64, No. 1, 61–72 (1976).
[2] J. Dieudonne,Foundations of Modern Analysis, Academic Press, New York, 1969.
[3] R. Hermann and A. J. Krener, Non linear controllability and observability,IEEE Trans. Automat. Control., AC 22, No. 5, 728–740 (1977). · Zbl 0396.93015
[4] L. R. Hunt, Controllability of general linear systems,Math. System Theory, 12, 361–370 (1979). · Zbl 0408.93011
[5] L. R. Hunt, Global controllability of non linear systems in two dimensions,Math. Systems Theory, 13, 361–376 (1980). · Zbl 0446.93016
[6] L. R. Hunt, Controllability of non linear hypersurface systems, in: Algebtraic and geometric methods in linear systems theory,AMS Lectures in Applied Mathematics, 18, C. I. Byrnes and C. F. Martin, Eds., pp. 209–224.
[7] L. R. Hunt,n-dimensional controllability with (n) controls,IEEE Trans. Automat. Control, AC 27, No. 1, 113–117 (1982). · Zbl 0476.93017
[8] V. Jurdjevic, Attainable sets and controllability; a geometric approach, in:Lectures Notes in Econom. and Math. Systems, No. 106, pp. 219–251. · Zbl 0324.93009
[9] V. Jurdjevic and I. Kupka, Control systems on semi-simple Lie groups and their homogeneous spaces,Ann. Inst. Fourier, 31, 4, 151–179 (1981). · Zbl 0453.93011
[10] V. Jurdjevic and I. Kupka, Control systems subordinated to a group action: Accessibility,J. Differential Equations, 39, No. 2, 186–211 (1981). · Zbl 0531.93008
[11] H. Kunita, On the controllability of non-linear systems,Appl. Math Optim., 5, 89–99 (1979). · Zbl 0406.93011
[12] I. Kupka and G. Sallet, A sufficient condition for the transitivity of pseudo-semi-groups. Application to system theory,J. Differential Equations, to appear. · Zbl 0527.93029
[13] C. Lobry, Controlabilité des systèmes non lineaires,SIAM J. Control, 8, 573–605 (1970). · Zbl 0207.15201
[14] C. Lobry, Bases mathématiques de la théorie du contrôle, cours de troisième cycle. Multigraphié, Bordeaux (1978).
[15] C. Lobry and P. Brunovsky, Controlabilité Bang-Bang, controlabilité differentiable, et perturbations des systèmes linéaires,Ann. Mat. Appl., sér. 4, 55, 93–119 (1975). · Zbl 0316.93007
[16] H. J. Sussmann, Orbits of families of vector fields and integrability of distributions,Trans. Amer. Math. Soc., 180, 171–188 (1973). · Zbl 0274.58002
[17] H. J. Sussmann and V. Jurdjevic, Controllability of nonlinear systems,J. Differential Equations, 00, 95–116 (1972). · Zbl 0242.49040
[18] A. Bacciotti and G. Stephani, The region of attainability of nonlinear system with unbounded controls,J. Optimization Theory and Applications, 35, 1, 57–84 (1981). · Zbl 0441.93019
[19] A. Bacciotti and G. Stephani, On the relationship between global and local controllability,Math. Systems Theory, 16, 79–91 (1983). · Zbl 0506.93007
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