We use a method of investigation based on employing adequate variational calculus techniques in the study of some generalized Dieudonné-Rashevski problems. This approach allows us to state and prove optimality conditions for such kind of vector multitime variational problems, with mixed isoperimetric constraints. We state and prove efficiency conditions and develop a duality theory.
References
K. J. Åström and R. M. Murray, Feedback Systems: An Introduction for Scientists and Engineers, Princeton University Press, Princeton, NJ, USA, 2008. MR2400446 K. J. Åström and R. M. Murray, Feedback Systems: An Introduction for Scientists and Engineers, Princeton University Press, Princeton, NJ, USA, 2008. MR2400446
C. Udrişte, “Multitime controllability, observability and bang-bang principle,” Journal of Optimization Theory and Applications, vol. 139, no. 1, pp. 141–157, 2008. 1156.93013 MR2438599 10.1007/s10957-008-9430-2 C. Udrişte, “Multitime controllability, observability and bang-bang principle,” Journal of Optimization Theory and Applications, vol. 139, no. 1, pp. 141–157, 2008. 1156.93013 MR2438599 10.1007/s10957-008-9430-2
C. Udrişte and I. $\c{T}$evy, “Multitime dynamic programming for curvilinear integral actions,” Journal of Optimization Theory and Applications, vol. 146, no. 1, pp. 189–207, 2010. 1202.49027 MR2657831 10.1007/s10957-010-9664-7 C. Udrişte and I. $\c{T}$evy, “Multitime dynamic programming for curvilinear integral actions,” Journal of Optimization Theory and Applications, vol. 146, no. 1, pp. 189–207, 2010. 1202.49027 MR2657831 10.1007/s10957-010-9664-7
Ş. Mititelu, “Optimality and duality for invex multi-time control problems with mixed constraints,” Journal of Advanced Mathematical Studies, vol. 2, no. 1, pp. 25–34, 2009. 1177.49038 MR2542703 Ş. Mititelu, “Optimality and duality for invex multi-time control problems with mixed constraints,” Journal of Advanced Mathematical Studies, vol. 2, no. 1, pp. 25–34, 2009. 1177.49038 MR2542703
Ş. Mititelu, A. Pitea, and M. Postolache, “On a class of multitime variational problems with isoperimetric constraints,” Scientific Bulletin, Series A, vol. 72, no. 3, pp. 31–40, 2010. MR2724462 1249.90260 Ş. Mititelu, A. Pitea, and M. Postolache, “On a class of multitime variational problems with isoperimetric constraints,” Scientific Bulletin, Series A, vol. 72, no. 3, pp. 31–40, 2010. MR2724462 1249.90260
A. Pitea and M. Postolache, “Minimization of vectors čommentComment on ref. [11?]: Please update the information of these references [11, 12, 13?], if possible. of curvilinear functionals on the second order jet bundle Sufficient efficiency conditions,” Optimization Letters. In press. A. Pitea and M. Postolache, “Minimization of vectors čommentComment on ref. [11?]: Please update the information of these references [11, 12, 13?], if possible. of curvilinear functionals on the second order jet bundle Sufficient efficiency conditions,” Optimization Letters. In press.
A. Pitea and M. Postolache, “Minimization of vectors of curvilinear functionals on the second order jet bundle. Necessary conditions,” Optimization Letters, vol. 6, no. 3, pp. 459–470, 2012. MR2891684 06111668 10.1007/s11590-010-0272-0 A. Pitea and M. Postolache, “Minimization of vectors of curvilinear functionals on the second order jet bundle. Necessary conditions,” Optimization Letters, vol. 6, no. 3, pp. 459–470, 2012. MR2891684 06111668 10.1007/s11590-010-0272-0
A. Pitea and M. Postolache, “Duality theorems for a new class of multitime multiobjective variational problems,” Journal of Global Optimization. In press. A. Pitea and M. Postolache, “Duality theorems for a new class of multitime multiobjective variational problems,” Journal of Global Optimization. In press.
A. Pitea, C. Udrişte, and Ş. Mititelu, “New type dualities in PDI and PDE constrained optimization problems,” Journal of Advanced Mathematical Studies, vol. 2, no. 1, pp. 81–90, 2009. 1177.49054 MR2542709 A. Pitea, C. Udrişte, and Ş. Mititelu, “New type dualities in PDI and PDE constrained optimization problems,” Journal of Advanced Mathematical Studies, vol. 2, no. 1, pp. 81–90, 2009. 1177.49054 MR2542709
T. W. Ting, “Elastic-plastic torsion of convex cylindrical bars,” Journal of Mathematics and Mechanics, vol. 19, pp. 531–551, 1969. MR250546 0197.23301 T. W. Ting, “Elastic-plastic torsion of convex cylindrical bars,” Journal of Mathematics and Mechanics, vol. 19, pp. 531–551, 1969. MR250546 0197.23301
J. A. Andrejewa and R. Klötzler, “Zur analytischen Lösung geometrischer Optimierungsaufgaben mittels Dualität bei Steuerungsproblemen. I,” Zeitschrift für Angewandte Mathematik und Mechanik, vol. 64, no. 1, pp. 35–44, 1984. MR736678 J. A. Andrejewa and R. Klötzler, “Zur analytischen Lösung geometrischer Optimierungsaufgaben mittels Dualität bei Steuerungsproblemen. I,” Zeitschrift für Angewandte Mathematik und Mechanik, vol. 64, no. 1, pp. 35–44, 1984. MR736678
J. A. Andrejewa and R. Klötzler, “Zur analytischen Lösung geometrischer Optimierungsaufgaben mittels Dualität bei Steuerungsproblemen. II,” Zeitschrift für Angewandte Mathematik und Mechanik, vol. 64, no. 3, pp. 147–153, 1984. MR748301 J. A. Andrejewa and R. Klötzler, “Zur analytischen Lösung geometrischer Optimierungsaufgaben mittels Dualität bei Steuerungsproblemen. II,” Zeitschrift für Angewandte Mathematik und Mechanik, vol. 64, no. 3, pp. 147–153, 1984. MR748301
B. Mond and I. Smart, “Duality and sufficiency in control problems with invexity,” Journal of Mathematical Analysis and Applications, vol. 136, no. 1, pp. 325–333, 1988. 0667.49001 MR972603 10.1016/0022-247X(88)90135-7 B. Mond and I. Smart, “Duality and sufficiency in control problems with invexity,” Journal of Mathematical Analysis and Applications, vol. 136, no. 1, pp. 325–333, 1988. 0667.49001 MR972603 10.1016/0022-247X(88)90135-7
V. Preda, “On duality and sufficiency in control problems with general invexity,” Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie, vol. 35, no. 3-4, pp. 271–280, 1991. 0828.49021 MR1307693 V. Preda, “On duality and sufficiency in control problems with general invexity,” Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie, vol. 35, no. 3-4, pp. 271–280, 1991. 0828.49021 MR1307693
C. Udrişte, O. Dogaru, and I. $\c{T}$evy, “Null Lagrangian forms and Euler-Lagrange PDEs,” Journal of Advanced Mathematical Studies, vol. 1, no. 1-2, pp. 143–156, 2008. 1169.49022 MR2498896 C. Udrişte, O. Dogaru, and I. $\c{T}$evy, “Null Lagrangian forms and Euler-Lagrange PDEs,” Journal of Advanced Mathematical Studies, vol. 1, no. 1-2, pp. 143–156, 2008. 1169.49022 MR2498896
L. D. Berkovitz, “Variational methods in problems of control and programming,” Journal of Mathematical Analysis and Applications, vol. 3, pp. 145–169, 1961. 0100.31005 MR139030 10.1016/0022-247X(61)90013-0 L. D. Berkovitz, “Variational methods in problems of control and programming,” Journal of Mathematical Analysis and Applications, vol. 3, pp. 145–169, 1961. 0100.31005 MR139030 10.1016/0022-247X(61)90013-0
Ş. Mititelu, V. Preda, and M. Postolache, “Duality of multitime vector integral programming with quasiinvexity,” Journal of Advanced Mathematical Studies, vol. 4, no. 2, pp. 59–72, 2011. MR2856578 1239.65040 Ş. Mititelu, V. Preda, and M. Postolache, “Duality of multitime vector integral programming with quasiinvexity,” Journal of Advanced Mathematical Studies, vol. 4, no. 2, pp. 59–72, 2011. MR2856578 1239.65040
M. A. Noor, “Nonconvex quasi variational inequalities,” Journal of Advanced Mathematical Studies, vol. 3, no. 1, pp. 59–72, 2010. 1206.49011 MR2606118 M. A. Noor, “Nonconvex quasi variational inequalities,” Journal of Advanced Mathematical Studies, vol. 3, no. 1, pp. 59–72, 2010. 1206.49011 MR2606118
M. Postolache, “Minimization of vectors of curvilinear functionals on 2nd order jet bundle: dual program theory,” Abstract and Applied Analysis, vol. 2012, Article ID 535416, 2012. MR2898036 M. Postolache, “Minimization of vectors of curvilinear functionals on 2nd order jet bundle: dual program theory,” Abstract and Applied Analysis, vol. 2012, Article ID 535416, 2012. MR2898036
P. Wolfe, “A duality theorem for non-linear programming,” Quarterly of Applied Mathematics, vol. 19, pp. 239–244, 1961. 0109.38406 MR135625 P. Wolfe, “A duality theorem for non-linear programming,” Quarterly of Applied Mathematics, vol. 19, pp. 239–244, 1961. 0109.38406 MR135625