Smooth and Nonsmooth Optimal Lipschitz Control — a Model Problem

  • Manfred Goebel
Part of the International Series of Numerical Mathematics book series (ISNM, volume 124)


For a semilinear second order ordinary differential equation a control problem with pointwise state constraints is considered. The control of the system is realized via the outer function of the nonlinear term in the state equation. We prove a simple theorem stating the existence of optimal controls within the class of Lipschitz continuous functions and derive two necessary optimality conditions depending on their smoothness properties. For a smooth optimal control function the given optimality condition looks somehow like Pontryagin’s maximum principle. In the general case the optimality condition involves Clarke’s generalized directional derivative. Furthermore a modified problem is considered, for which by means of Ekeland’s variational principle a suboptimality condition can be proved.


Optimal Control Problem Admissible Control Lipschitz Continuous Function Order Ordinary Differential Equation Adjoint State 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Kluge, R.: Zur Parameterbestimmung in nichtlinearen Problemen, volume 81 of Teubner-Texte zur Mathematik. BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1985.Google Scholar
  2. [2]
    Rösch, A.: Identifikation nichtlinearer Wärmeübergangsgesetze mit Methoden der Optimalen Steuerung. Diss., Techn. Univ. Chemnitz-Zwickau, 1995.zbMATHGoogle Scholar
  3. [3]
    Barbu, V.; Kunisch, K.: Identification of Nonlinear Elliptic Equations. Appl. Math. Optim., 33, 139–167, 1996.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    Barbu, V.; Kunisch, K.: Identification of Nonlinear Parabolic Equations. Preprint, Techn. Univ. Graz, Inst. Math., 1996.Google Scholar
  5. [5]
    Goebel, M.; Oestreich, D.: Optimal Control of a Nonlinear Singular Integral Equation Arising in Electrochemical Machining. Z. Anal. Anwend., 10(1), 73–82, 1991.MathSciNetzbMATHGoogle Scholar
  6. [6]
    Recknagel, G.: Zur Lösung und optimalen Steuerung einer gewöhnlichen nichtlinearen Randwertaufgabe zweiter Ordnung. Diplomarb., Martin-Luther-Univ. Halle-Wittenberg, FB Math./Inf., 1995.Google Scholar
  7. [7]
    Goebel, M.: Smooth and Nonsmooth Optimal Lipschitz Control, manuscript to be published.Google Scholar
  8. [8]
    Schleiff, M.: Eindimensionale elektrothermische Prozesse und Randwertaufgaben für eine Differentialgleichung 2. Ordnung. Z. Angew. Math. Mech., 66(10), 483–488, 1986.zbMATHCrossRefGoogle Scholar
  9. [9]
    Clarke, F. H.: Optimization and nonsmooth analysis. SIAM, Philadelphia, 1990.zbMATHCrossRefGoogle Scholar
  10. [10]
    Mäkelä, M. M.; Neittaanmäki, P.: Nonsmooth optimization. World Scientific Publishing Co., Singapore, New Jersey, London, Hong Kong, 1992.zbMATHGoogle Scholar
  11. [11]
    Ekeland, I.: Nonconvex Minimization Problems. Bull. Am. Math. Soc. (N. S.), 1(3), 443–474, 1979.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 1998

Authors and Affiliations

  • Manfred Goebel
    • 1
  1. 1.FB Mathematik und InformatikMartin-Luther-Universität Halle-WittenbergHalle (Saale)Germany

Personalised recommendations