Controllability properties of nonlinear behaviors

Colonius, Fritz; Kliemann, Wolfgang

doi:10.1090/S0002-9947-08-04612-6

Controllability properties of nonlinear behaviors
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by Fritz Colonius and Wolfgang Kliemann PDF

Trans. Amer. Math. Soc. 360 (2008), 5667-5682 Request permission

Abstract:

This paper proposes a topological framework for the analysis of the time shift on behaviors. It is shown that controllability is not a property of the time shift, while chain controllability is. This also leads to a global decomposition of behaviors.

References

M. V. Bebutov, Dynamical systems in the space of continuous functions, Dokl. Akad. Nauk SSSR, 27 (1940), pp. 904–906. In Russian.
Fritz Colonius and Wolfgang Kliemann, Some aspects of control systems as dynamical systems, J. Dynam. Differential Equations 5 (1993), no. 3, 469–494. MR 1235039, DOI 10.1007/BF01053532
—, The Dynamics of Control, Birkhäuser, Boston, 2000.
—, Behaviors and controllability, in Electronic Proceedings of the 16th Conference on Mathematical Theory of Networks and Systems (MTNS), July 5-9, 2004, Leuven (Belgium), 2004.
Fritz Colonius and Marco Spadini, Uniqueness of local control sets, J. Dynam. Control Systems 9 (2003), no. 4, 513–530. MR 2001957, DOI 10.1023/A:1025692218217
Robert W. Easton, Geometric methods for discrete dynamical systems, Oxford Engineering Science Series, vol. 50, Oxford University Press, New York, 1998. MR 1616726
T. Gayer, Control sets and their boundaries under parameter variation, J. Differential Equations 201 (2004), no. 1, 177–200. MR 2057543, DOI 10.1016/j.jde.2004.02.006
Martin Golubitsky and David G. Schaeffer, Singularities and groups in bifurcation theory. Vol. I, Applied Mathematical Sciences, vol. 51, Springer-Verlag, New York, 1985. MR 771477, DOI 10.1007/978-1-4612-5034-0
Sergei Yu. Pilyugin, The space of dynamical systems with the $C^0$-topology, Lecture Notes in Mathematics, vol. 1571, Springer-Verlag, Berlin, 1994. MR 1329732, DOI 10.1007/BFb0073519
Jan Willem Polderman and Jan C. Willems, Introduction to mathematical systems theory, Texts in Applied Mathematics, vol. 26, Springer-Verlag, New York, 1998. A behavioral approach. MR 1480665, DOI 10.1007/978-1-4757-2953-5
Aubrey B. Poore, A model equation arising from chemical reactor theory, Arch. Rational Mech. Anal. 52 (1973), 358–388. MR 338508, DOI 10.1007/BF00247470
Clark Robinson, Dynamical systems, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. Stability, symbolic dynamics, and chaos. MR 1396532
Dietmar Salamon and Eduard Zehnder, Flows on vector bundles and hyperbolic sets, Trans. Amer. Math. Soc. 306 (1988), no. 2, 623–649. MR 933310, DOI 10.1090/S0002-9947-1988-0933310-9
N. V. Ščerbina, The continuity of one-parameter families of sets, Dokl. Akad. Nauk SSSR 234 (1977), no. 2, 327–329 (Russian). MR 0440500
George R. Sell, Topological dynamics and ordinary differential equations, Van Nostrand Reinhold Mathematical Studies, No. 33, Van Nostrand Reinhold Co., London, 1971. MR 0442908
Dietmar Szolnoki, Set oriented methods for computing reachable sets and control sets, Discrete Contin. Dyn. Syst. Ser. B 3 (2003), no. 3, 361–382. MR 1974152, DOI 10.3934/dcdsb.2003.3.361
Jan C. Willems, Models for dynamics, Dynamics reported, Vol. 2, Dynam. Report. Ser. Dynam. Systems Appl., vol. 2, Wiley, Chichester, 1989, pp. 171–269. MR 1000978