Variational problems within the class of solutions of a partial differential equation
HTML articles powered by AMS MathViewer
- by Robert Delver PDF
- Trans. Amer. Math. Soc. 180 (1973), 265-289 Request permission
Abstract:The subject of this paper is the optimization of a multiple integral over a domain $G$ of a function, containing as arguments the independent variables, the unknown function and its partial derivatives up to order $l$, within the class of all sufficiently smooth solutions in $G$ of a given partial differential equation of order greater than or equal to $2l$. Necessary conditions in the form of a boundary value problem are derived. A physical application occurs in the control with boundary and initial conditions of a process in $G$ that is described by a specific partial differential equation.
- R. Delver, The optimization of a functional within the class of solutions of a partial differential equation, J. Engrg. Math. 3 (1969), 253–264. MR 257848, DOI 10.1007/BF01535298 —, Variational problems within the class of solutions of a partial differential equation, Ph. D. Thesis, University of California, Berkeley, Calif., 1971.
- Avner Friedman, Partial differential equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. MR 0445088 —, Optimal control for parabolic equations, J. Math. Anal. Appl. 18 (1967), 479-491.
- K. A. Lurie, The Mayer-Bolza problem for multiple integrals: Some optimum problems for elliptic differential equations arising in magnetohydrodynamics, Topics in Optimization, Academic Press, New York, 1967, pp. 147–193. MR 0210487 J. L. Lions, Controle optimal de systèmes gouvernés par des équations aux dérivées partielles, Dunod; Gauthier-Villars, Paris, 1968; English transl., Die Grundlehren der math. Wissenschaften, Band 170, Springer-Verlag, Berlin and New York, 1971. MR 39 #5920; MR 42 #6395.
- Carlo Miranda, Equazioni alle derivate parziali di tipo ellittico, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Heft 2, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1955 (Italian). MR 0087853
- Hassler Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), no. 1, 63–89. MR 1501735, DOI 10.1090/S0002-9947-1934-1501735-3
- Robert Delver, Variational problems within the class of solutions of a partial differential equation, Bull. Amer. Math. Soc. 77 (1971), 1100–1105. MR 284902, DOI 10.1090/S0002-9904-1971-12884-7 —, Boundary and interior control for partial differential equations (to appear).
- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 180 (1973), 265-289
- MSC: Primary 49B15
- DOI: https://doi.org/10.1090/S0002-9947-1973-0320856-9
- MathSciNet review: 0320856