Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Existence theorems for multidimensional Lagrange problems

  • 78 Accesses

  • 24 Citations

Abstract

Existence theorems are proved for multidimensional Lagrange problems of the calculus of variations and optimal control. The unknowns are functions of several independent variables in a fixed bounded domain, the cost functional is a multiple integral, and the side conditions are partial differential equations, not necessarily linear, with assigned boundary conditions. Also, unilateral constraints may be prescribed both on the space and the control variables. These constraints are expressed by requiring that space and control variables take their values in certain fixed or variable sets wich are assumed to be closed but not necessarily compact.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Cesari, L.,Existence Theorems for Multidimensional Problems of Optimal Control, Differential Equations and Dynamical Systems, pp 115–132, Academic Press, New York, 1967.

  2. 2.

    Cesari, L.,Existence Theorems for Weak and Usual Optimal Solutions in Lagrange Problems with Unilateral Constraints, I and II, Transactions of the American Mathematical Society, Vol. 124, pp. 369–412 and 413–429, 1966.

  3. 3.

    Cesari, L.,Existence Theorems for Optimal Solutions in Pontryagin and Lagrange Problems, SIAM Journal on Control, Vol. 3, pp. 475–498, 1965.

  4. 4.

    Sobolev, S. L.,Applications of Functional Analysis in Mathematical Physics, American Mathematical Society, Providence, Rhode Island, 1963.

  5. 5.

    Choquet, G.,Convergences, Annales de l'Université de Grenoble, Vol. 23, pp. 55–112, 1947–48.

  6. 6.

    Kuratowski, C., Les Fonctions Semi-Continues dans l'Espace des Ensembles Fermés, Fundamenta Mathematicae, Vol. 18, pp. 148–166, 1932.

  7. 7.

    Michael, E.,Topologies on Spaces of Subsets, Transactions of the American Mathematical Society, Vol. 71, pp. 152–182, 1951.

  8. 8.

    Riesz, F., andNagy, B. S.,Functional Analysis, Ungar, New York, 1955.

  9. 9.

    Castaing, C., Quelques Problèmes de Mesurabilité Liés à la Théorie de la Commande, Comptes Rendus de l'Académie des Sciences, Vol. 262, Série A, pp. 409–411, 1966.

Download references

Author information

Additional information

This research was partially supported by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-942-65.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Cesari, L. Existence theorems for multidimensional Lagrange problems. J Optim Theory Appl 1, 87–112 (1967). https://doi.org/10.1007/BF00936648

Download citation

Keywords

  • Boundary Condition
  • Differential Equation
  • Control Variable
  • Partial Differential Equation
  • Bounded Domain