Integral representations are obtained for solutions of a Darboux problem in a rectangle and used to prove Neustadt-type existence theorems for optimal control problems with trajectories satisfying linear, hyperbolic partial differential equations with Darboux-type boundary data. The proof bears on the fact that, in this situation, for each generalized solution, there is a usual solution where the functional takes the same value.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Neustadt, L.,The Existence of Optimal Control in the Absence of Convexity, Journal of Mathematical Analysis and Applications, Vol. 7, pp. 110–117, 1963.
Germany, R. H. J.,Sur les Fonctions de Riemann Associées aux Systèmes d'Equations aux Derivées Partielles et d'Equations Integro-Differentielles du Second Ordre a Deux Variables Independantes, Memoires de la Societé Royale des Sciences de Liège, Ser. 3, Vol. 14, Fasc. 4, pp. 1–66, 1928.
Cesari, L.,Optimization with Partial Differential Equations in Dieudonné-Rashevsky Form and Conjugate Problem, Archive for Rational Mechanics and Analysis, Vol. 33, pp. 339–357, 1969.
Suryanarayana, M. B.,Necessary Conditions for Optimization Problems with Hyperbolic Partial Differential Equations, SIAM Journal on Control, Vol. 11, No. 1, 1973.
Suryanarayana, M. B.,Existence Theorems for Optimization Problems Concerning Hyperbolic Partial Differential Equations, Journal of Optimization Theory and Applications, Vol. 15, No. 4, 1975.
Pulvirenti, G.,Existence Theorems for an Optimal Control Problem Relative to a Linear Hyperbolic Partial Differential Equation, Journal of Optimization Theory and Applications, Vol. 7, No. 2, 1971.
Suryanarayana, M. B.,Linear Control Problems with Total Differential Equations Without Convexity, Transactions of the American Mathematical Society, Vol. 200, pp. 233–249, 1974.
Suryanarayana, M. B.,Multidimensional Integral Equations of Volterra Type, Pacific Journal of Mathematics, Vol. 41, No. 3, 1972.
Gunning, R. C., andRossi, H.,Analytic Functions of Several Complex Variables, Prentice-Hall, Englewood Cliffs, New Jersey, 1965.
Sobolev, S. L.,Applications of Functional Analysis in Mathematical Physics, Vol. 7, Translations of Mathematical Monographs, American Mathematical Society, Providence, Rhode Island, 1963.
Blackwell, D.,The Range of Certain Vector Integrals, Proceedings of the American Mathematical Society, Vol. 2, pp. 390–395, 1951.
Olech, C.,Lexicographical Order, Range of Integrals, and Bang-Bang Principle, Mathematical Theory of Control, Edited by A. V. Balakrishnan and L. Neustadt, Academic Press, New York, New York, 1967.
Castaing, C.,Some Theorems in Measure Theory and Generalized Dynamic Systems Defined by Contingent Equations, Mathematical Systems Theory and Economics, II, Edited by H. W. Kuhn and G. P. Szego, Springer-Verlag, New York, New York, 1969.
Cesari, L.,An Existence Theorem without Convexity Conditions, SIAM Journal on Control, Vol. 12, No. 2, 1974.
Cesari, L.,Problems of Optimization, Vols. I and II, Springer-Verlag, New York, New York (to appear).
Jacobs, M. Q.,Attainable Sets of Linear Systems with Unbounded Controls, Mathematical Theory of Control, Edited by A. V. Balakrishnan and L. Neustadt, Academic Press, New York, New York, 1967.
Cesari, L.,Convexity of the Range of Certain integrals, SIAM Journal on Control, Vol. 13, No. 2, 1975.
McShane, E. J., andWarfield, R. B.,On Filippov's Implicit Function Lemma, Proceedings of the American Mathematical Society, Vol. 18, pp. 41–47, 1967.
This work was done in the framework of Research Project AFOSR-69-1662. The author is greatly indebted to Professor L. Cesari for his valuable guidance and constant encouragement during the writing of this paper.
Communicated by L. Cesari
About this article
Cite this article
Suryanarayana, M.B. Existence theorems for optimization problems concerning linear, hyperbolic partial differential equations without convexity conditions. J Optim Theory Appl 19, 47–61 (1976). https://doi.org/10.1007/BF00934051
- Existence theorems
- linear systems
- Green's functions
- hyperbolic partial differential equations
- multidimensional control problems
- relaxed solutions