Advertisement

Exact Bang-Bang Optimal Control for Problems with Nonlinear Costs

  • Erik J. Balder
Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 382)

Abstract

In [5] a general functional-analytic theory of “existence without convexity” was developed and applied to control and variational problems. Apart from being fundamental, this theory also leads to an extension of classical existence results in very concrete cases, because of its systematic incorporation of concavity — instead of classical linearity — for the trajectory cost function. Here the theory is shown to apply also to bang-bang optimal control, for which, likewise, (i) a comprehensive treatment is given and (ii) all classical results are shown to be extendible by the incorporation of a concave trajectory cost function.

Keywords

Lower Semicontinuous Control Pair Lower Semi Abstract Existence Linear Integral Operator 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    T. Angell, Existence of optimal control without convexity and a bang-bang theorem for linear Volterra equations, J. Optim. Theory Appl. 19 (1976), 63–79.CrossRefGoogle Scholar
  2. [2]
    E.J. Balder, Existence results by extremal properties of the original controls, unpublished preprint, 1978.Google Scholar
  3. [3]
    E.J. Balder, An extension of the usual model in statistical decision theory with applications to stochastic optimization problems (abstract), Adv. Appl. Probab. 10 (1978), 325–326.CrossRefGoogle Scholar
  4. [4]
    E.J. Balder, New sequential compactness results for spaces of scalarly integrable functions, J. Math. Anal. Appl 151 (1990), 1–16.CrossRefGoogle Scholar
  5. [5]
    E.J. Balder, New existence results for optimal controls in the absence of convexity: the importance of extremality, forthcoming.Google Scholar
  6. [6]
    H. Berliocchi and J.-M. Lasry, Intégrandes normales et mesures paramétrées en calcul des variations, Bull. Soc. Math. France 101 (1973), 129–184.Google Scholar
  7. [7]
    C. Castaing and M. Valadier, Convex Analysis and Measurable Multifonctions, Lecture Notes in Mathematics 580, Springer-Verlag, Berlin, 1977.CrossRefGoogle Scholar
  8. [8]
    A. Cellina and G. Colombo, On a classical problem of the calculus of variations without convexity assumptions, Ann. Inst. H. Poincaré, Analyse Nonlinéaire 7 (1989), 97–106.Google Scholar
  9. [9]
    L. Cesari, Optimization Theory and Applications: Problems with Ordinary Differential Equations, Springer-Verlag, Berlin, 1983.Google Scholar
  10. [10]
    G. Choquet, Lectures on Analysis, Benjamin, Reading, Mass., 1969.Google Scholar
  11. [11]
    J. Diestel, Remarks on weak compactness in L 1 (μ, X), Glasgow Math. J. 18 (1977), 87–91.CrossRefGoogle Scholar
  12. [12]
    H. Hermes and J.P. Lasalle, Functional Analysis and Time Optimal Control, Academic Press, New York, 1969.Google Scholar
  13. [13]
    R.K. Miller, Nonlinear Volterra Integral Equations, Benjamin, Menlo Park, California, 1971.Google Scholar
  14. [14]
    L.W. Neustadt, The existence of optimal controls in the absence of convexity conditions, J. Math. Anal. Appl. 7 (1963), 110–117.CrossRefGoogle Scholar
  15. [15]
    J.-P. Raymond, Conditions nécessaires et suffisantes d’existence de solutions en calcul des variations, Ann. Inst. H. Poincaré, Analyse non linéaire 4 (1987), 169–202.Google Scholar
  16. [16]
    J.-P. Raymond, Existence theorems in optimal control theory without convexity assumptions, J. Optim. Theory Appl. 67 (1990), 109–132.CrossRefGoogle Scholar
  17. [17]
    J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Erik J. Balder
    • 1
  1. 1.Mathematical InstituteUniversity of UtrechtUtrechtthe Netherlands

Personalised recommendations