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Convexity of reachable sets of nonlinear ordinary differential equations

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Abstract

A necessary and sufficient condition for the reachable set, i.e., the set of states reachable from a ball of initial states at some time, of an ordinary differential equation to be convex is presented. In particular, convexity is guaranteed if the ball of initial states is sufficiently small, an upper bound on the radius of that ball being obtained directly from the right hand side of the differential equation. In finite dimensions, the results cover the case of ellipsoids of initial states. A potential application of the results is inner and outer polyhedral approximation of reachable sets, which becomes extremely simple and almost universally applicable if these sets are known to be convex. An example demonstrates that the balls of initial states for which the latter property follows from the results are large enough to be used in actual computations.

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References

  1. Hsu, C.S., Cell-to-Cell Mapping, vol. 64 of Applied Mathematical Sciences, New York: Springer, 1987.

    Google Scholar 

  2. Blanke, M., Kinnaert, M., Lunze, J., and Staroswiecki, M., Diagnosis and Fault-Tolerant Control, Berlin: Springer, 2003.

    MATH  Google Scholar 

  3. Chutinan, A. and Krogh, B.H., Computational Techniques for Hybrid System Verification, IEEE Trans. Automat. Control, 2003, vol. 48, no. 1, pp. 64–75.

    Article  MathSciNet  Google Scholar 

  4. Tomlin, C.J., Mitchell, I., Bayen, A.M., and Oishi, M., Computational Techniques for the Verification of Hybrid Systems, Proc. IEEE, 2003, vol. 91, no. 7, pp. 986–1001.

    Article  Google Scholar 

  5. Kurzhanski, A.B. and Varaiya, P., On Verification of Controlled Hybrid Dynamics through Ellipsoidal Techniques, Proc. 44th IEEE Conf. Decision and Control Eur. Control Conf., Seville, 2005, pp. 4682–4687.

  6. Singer, A.B. and Barton, P.I., Global Optimization with Nonlinear Ordinary Differential Equations, J. Global Optim., 2006, vol. 34, no. 2, pp. 159–190.

    Article  MATH  MathSciNet  Google Scholar 

  7. Junge, O. and Osinga, H.M., A Set Oriented Approach to Global Optimal Control, ESAIM Control Optim. Calc. Var., 2004, vol. 10, no. 2, pp. 259–270 (electronic).

    Article  MATH  MathSciNet  Google Scholar 

  8. Hwang, I., Stipanović, D.M., and Tomlin, C.J., Polytopic Approximations of Reachable Sets Applied to Linear Dynamic Games and a Class of Nonlinear Systems, in Advances in Control, Communication Networks, and Transportation Systems, Syst. Control Found. Appl., Boston: Birkhäuser, 2005, pp. 3–19.

    Google Scholar 

  9. Hartman, P., Ordinary Differential Equations, vol. 38 of Classics in Applied Mathematics, Philadelphia: SIAM, 2002.

    Google Scholar 

  10. Veliov, V., On the Time-Discretization of Control Systems, SIAM J. Control Optim., 1997, vol. 35, no. 5, pp. 1470–1486.

    Article  MATH  MathSciNet  Google Scholar 

  11. Baturin, V.A., Goncharova, E.V., Pereĭra, F.L., and Souza, Z.B., Polyhedral Approxomations of the Boundary of the Attainable Set of Measure-Controlled Dynamical Systems, Avtom. Telemekh., 2006, no. 3, pp. 8–19.

  12. Berkovitz, L.D., Optimal Control Theory, vol. 12 of Applied Mathematical Sciences, New York: Springer, 1974.

    Google Scholar 

  13. Macki, J.W. and Strauss, A., Introduction to Optimal Control Theory, New York: Springer, 1982.

    MATH  Google Scholar 

  14. Giannessi, F., Constrained Optimization and Image Space Analysis, vol. 1: Separation of Sets and Optimality Conditions, vol. 49 of Mathematical Concepts and Methods in Science and Engineering, New York: Springer, 2005

    Google Scholar 

  15. Hadamard, J., Sur certaines propriétés des trajectoires en dynamique, Journ. de Math, 1897, vol. 5, no. 3, pp. 331–387.

    Google Scholar 

  16. Toeplitz, O., Das algebraische Analogon zu einem Satze von Fejér, Math. Z., 1918, vol. 2, no. 1–2, pp. 187–197.

    Article  MathSciNet  Google Scholar 

  17. Hausdorff, F., Der Wertvorrat einer Bilinearform, Math. Z., 1919, vol. 3, no. 1, pp. 314–316.

    Article  MathSciNet  Google Scholar 

  18. Zampieri, G. and Gorni, G. Local Homeo-and Diffeomorphisms: Invertibility and Convex Image, Bull. Austral. Math. Soc., 1994, vol. 49, no. 3, pp. 377–398.

    Article  MATH  MathSciNet  Google Scholar 

  19. Polyak, B.T., Local Programming, Zh. Vych. Mat. Mat. Phys., 2001, vol. 41, no. 9, pp. 1324–1331.

    MathSciNet  Google Scholar 

  20. Bobylev, N.A., Emel’yanov, S.V., and Korovin, S.K., Convexity of Images of Convex Sets under Smooth Maps, Nelineinaya Dinamika i Upravlenie, 2002, no. 2, pp. 23–32.

  21. Vakhrameev, S.A., A Shift Formula and Its Applications to Some Smooth Nonlinear Controlled Systems, Sovrem. Mat. Prilozh., Ser.: Din. Sist, 2003, no. 4, pp. 117–139.

  22. Pliś, A., Accessible Sets in Control Theory, Proc. Int. Conf. Diff. Equat., Antosiewicz, H.A., Ed., New York: Academic, 1975.

    Google Scholar 

  23. Lojasiewicz, S., Jr., The Sufficiency of Maximum Principle, Proc. Conf. Analyse des systèmes (Bordeaux, 1978), Paris: Soc. Math. France, 1980, vol. 75–76 of Astérisque, pp. 187–197.

    Google Scholar 

  24. Polyak, B.T., Convexity of the Reachable Set of Nonlinear Systems under L 2 Bounded Controls, Dynam. Contin. Discrete Impuls. Syst. Ser. A Math. Anal, 2004, vol. 11, no. 2–3, pp. 255–267.

    MATH  MathSciNet  Google Scholar 

  25. Azhmyakov, V., Flockerzi, D., and Raisch, J., On Convexity of Reachable Sets for Nonlinear Control Systems, Proc. ECC, Kos, Grece, 2007.

  26. Hermes, H., On the Closure and Convexity of Attainable Sets in Finite and Infinite Dimensions, SIAM J. Control, 1967, vol. 5, pp. 409–417. Erratum: 1968, vol. 6, pp. 594–595.

    Article  MATH  MathSciNet  Google Scholar 

  27. Łojasiewicz, S., Jr., Some Properties of Accessible Sets in Nonlinear Control Systems, Ann. Polon. Math., 1979, vol. 36, no. 2, pp. 123–137.

    MathSciNet  MATH  Google Scholar 

  28. Topunov, M.V., On the Convexity of the Reachable Set of a Bilinear Control System, Prikl. Mat. Mekh., 2003, vol. 67, no. 5, pp. 752–758.

    MATH  MathSciNet  Google Scholar 

  29. Geist, S., Reißig, G., and Raisch, J., An Approach to the Computation of Reachable Sets of Nonlinear Dynamic Systems—An Important Step in Generating Discrete Abstractions of Continuous Systems, Proc. 11th IEEE Int. Conf. Methods and Models in Automation and Robotics (MMAR), Miedzyzdroje, Poland, 2005, pp. 101–106.

  30. Reißig, G. and Geist, S., Zur Kovexität erreichbarer Mengen nichtlinearer gewöhnlicher Differetialgleichungen, Tagungsband eines Workshops am Bostalsee (Saarland), Lohmann, B., Ed., pp. 302–314.

  31. Reißig, G., Convexity Criteria and Their Applications, E.S. Pyatnitskiy IX Workshop on Stability and Oscillations of Nonlinear Control Systems, Moscow, 2006.

  32. Muller, R.S. and Kamins, T.I., Device Electronics for Integrated Circuits, New York: Wiley, 1986.

    Google Scholar 

  33. Clarke, F.H., Ledyaev, Y.S., Stern, R.J., and Wolenski, P.R., Nonsmooth Analysis and Control Theory, vol. 178 of Graduate Texts in Mathematics, New York: Springer, 1998.

    Google Scholar 

  34. Pastor, K., Convexity and Generalized Second-Order Derivatives for Locally Lipschitz Functions, Nonlinear Anal., 2005, vol. 60, no. 3, pp. 547–555.

    Article  MATH  MathSciNet  Google Scholar 

  35. Sansone, G. and Conti, R., Non-Linear Differential Equations, vol. 67 of International Series of Monographs in Pure and Applied Mathematics, New York: Pergamon, 1964.

    MATH  Google Scholar 

  36. Delfour, M.C. and Zolésio, J.-P., Oriented Distance Function and Its Evolution Equation for Initial Sets with Thin Boundary, SIAM J. Control Optim., 2004, vol. 42, no. 6, pp. 2286–2304 (electronic).

    Article  MATH  MathSciNet  Google Scholar 

  37. Thorpe, J.A., Elementary Topics in Differential Geometry, in Undergraduate Texts in Mathematics, New York: Springer, 1979.

    Google Scholar 

  38. Polyak, B.T., Convexity of Nonlinear Image of a Small Ball with Applications to Optimisation, Set-Valued Anal., 2001, vol. 9, no. 1–2, pp. 159–168.

    Article  MATH  MathSciNet  Google Scholar 

  39. Kobayashi, S. and Nomizu, K., Foundations of Differential Geometry, New York: Interscience Publisher (Wiley), 1963, vol. 1.

    MATH  Google Scholar 

  40. Polyak, B.T., The Convexity Principle and Its Application, Bull. Braz. Math. Soc. (N.S.), 2003, vol. 34, no. 1, pp. 59–75.

    Article  MATH  MathSciNet  Google Scholar 

  41. Baldissera, F.L., Application of Hybrid System’s Approach to the Swing-Up Problem of an Inverted Pendulum, DAS 5511: Projeto de Film de Curso, Departamento de Automação e Sistemas, Centro Tecnológico, Universidade Federal de Santa Catarina, Florianópolis, Brasil, 2004.

    Google Scholar 

  42. Giles, J.R., Convex Analysis with Application in the Differentiation of Convex Functions, vol. 58 of Research Notes in Mathematics, Boston: Pitman, 1982.

    MATH  Google Scholar 

  43. Valentine, F.A., Convex Sets, in McGraw-Hill Series in Higher Mathematics, New York: McGraw-Hill, 1964.

    Google Scholar 

  44. Angeli, D. and Sontag, E.D., Monotone Control Systems, IEEE Trans. Automat. Control, 2003, vol. 48, no. 10, pp. 1684–1698.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Technische Universität Berlin, Berlin, Germany

    G. Reißig

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Original Russian Text © G. Reißig, 2007, published in Avtomatika i Telemekhanika, 2007, No. 9, pp. 64–78.

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Reißig, G. Convexity of reachable sets of nonlinear ordinary differential equations. Autom Remote Control 68, 1527–1543 (2007). https://doi.org/10.1134/S000511790709007X

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