A Young Measures Approach to Averaging

  • Zvi Artstein
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 75)


Employing a fast time scale in the Averaging Method results in a limit dynamics driven by a Young measure. The rate of convergence to the limit induces quantitative estimates for the averaging. Advantages that can be drawn from the Young measures approach, in particular, allowing time-varying averages, are displayed along with a connection to singularly perturbed systems.


Probability Measure Average Method Singular Perturbation Lipschitz Constant Classical Average 
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  1. [1]
    V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, second edition. Springer, New York, 1988.Google Scholar
  2. [2]
    Z. Artstein, Chattering variational limits of control systems. Forum Mathematicum 5 (1993), 369–403.MATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    Z. Artstein, On singularly perturbed ordinary differential equations with measure-valued limits. Mathematica Bohemica 127 (2002), 139–152.MATHMathSciNetGoogle Scholar
  4. [4]
    Z. Artstein, Distributional convergence in planar dynamics and singular perturbations. J. Differential Equations 201 (2004), 250–286.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Z. Artstein and A. Vigodner, Singularly perturbed ordinary differential equations with dynamic limits. Proceedings of the Royal Society of Edinburgh 126A (1996), 541–569MathSciNetGoogle Scholar
  6. [6]
    E. J. Balder, Lectures on Young measure theory and its applications to economics. Rend. Istit. Mat. Univ. Trieste 31 (2000), supplemento 1, 1–69.MATHMathSciNetGoogle Scholar
  7. [7]
    P. Billingsley, Convergence of Probability Measures. Wiley, New York, 1968.MATHGoogle Scholar
  8. [8]
    N.N. Bogoliubov and Y. A. Mitropolsky, Asymptotic Methods in the Theory of Non-linear Oscillations. English Translation, Gordon and Breach, New York, 1961.Google Scholar
  9. [9]
    J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Applied Mathematical Sciences 42. Springer-Verlag, New York, 1983.MATHGoogle Scholar
  10. [10]
    P. Lochak and C. Meunier, Multiphase Averaging for Classical Systems. With Applications to Adiabatic Theorems. Applied Mathematical Sciences, 72. Springer-Verlag, New York, 1988.MATHGoogle Scholar
  11. [11]
    R. E. O’Malley, Jr., Singular Perturbation Methods for Ordinary Differential Equations. Springer-Verlag, New York, 1991.MATHGoogle Scholar
  12. [12]
    P. Pedregal, Parameterized Measures and Variational Principles. Birkhäuser Verlag, Basel, 1997.Google Scholar
  13. [13]
    P. Pedregal, Optimization, relaxation and Young measures. Bull. Amer. Math. Soc. 36 (1999), 27–58.MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    J. A. Sanders and F. Verhulst, Averaging Methods in Nonlinear Dynamical Systems. Springer-Verlag, New York, 1985.MATHGoogle Scholar
  15. [15]
    A. N. Tikhonov, A.B. VasilÉva and A. G. Sveshnikov, Differential Equations. Springer-Verlag, Berlin, 1985.Google Scholar
  16. [16]
    M. Valadier, A course on Young measures. Rend. Istit. Mat. Univ. Trieste 26 (1994) supp., 349–394.MATHMathSciNetGoogle Scholar
  17. [17]
    F. Verhulst, Methods and Applications of Singular Perturbations. Texts in Applied Mathematics 50, Springer, New York, 2005.MATHGoogle Scholar
  18. [18]
    J. Warga, Optimal Control of Differential and Functional Equations. Academic Press, New York, 1972.MATHGoogle Scholar
  19. [19]
    L. C. Young, Lectures on the Calculus of Variations and Optimal Control Theory. Saunders, New York, 1969.MATHGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2007

Authors and Affiliations

  • Zvi Artstein
    • 1
  1. 1.Department of MathematicsThe Weizmann Institute of ScienceRehovotIsrael

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