Sufficient Conditions of Optimality for Convex Differential Inclusions of Elliptic Type and Duality
Author(s)
Elimhan N. Mahmudov
Affiliation(s)
Department of Industrial Engineering, Faculty of Management, Istanbul Technical University, Istanbul, Turkey.
Department of Industrial Engineering, Faculty of Management, Istanbul Technical University, Istanbul, Turkey.
ABSTRACT
This paper deals with the Dirichlet problem for convex differential (PC) inclusions of elliptic type. On the basis of conjugacy correspondence the dual problems are constructed. Using the new concepts of locally adjoint mappings in the form of Euler-Lagrange type inclusion is established extremal relations for primary and dual problems. Then duality problems are formulated for convex problems and duality theorems are proved. The results obtained are generalized to the multidimensional case with a second order elliptic operator.
This paper deals with the Dirichlet problem for convex differential (PC) inclusions of elliptic type. On the basis of conjugacy correspondence the dual problems are constructed. Using the new concepts of locally adjoint mappings in the form of Euler-Lagrange type inclusion is established extremal relations for primary and dual problems. Then duality problems are formulated for convex problems and duality theorems are proved. The results obtained are generalized to the multidimensional case with a second order elliptic operator.
Cite this paper
E. Mahmudov, "Sufficient Conditions of Optimality for Convex Differential Inclusions of Elliptic Type and Duality," Advances in Pure Mathematics, Vol. 2 No. 2, 2012, pp. 114-118. doi: 10.4236/apm.2012.22016.
E. Mahmudov, "Sufficient Conditions of Optimality for Convex Differential Inclusions of Elliptic Type and Duality," Advances in Pure Mathematics, Vol. 2 No. 2, 2012, pp. 114-118. doi: 10.4236/apm.2012.22016.
References
[1] A. Kurzhanski, “Set-Valued Analysis and Differential Inclusions,” Birkh?user, Boston, 1993. [2] J.-L. Lions, “Optimal Control of Systems Governed by Partial Differential Equations,” Springer, Berlin, 1991. [3] J. P. Aubin and A. Cellina, “Differential Inclusion,” Sp- ringer-Verlag, Berlin, 1984. [4] R. T. Rockafellar, “Convex Analysis,” 2nd Edition, Princeton University, Hoboken, 1972. [5] V. L. Makarov and A. M. Rubinov, “The Mathematical Theory of Economic Dynamics and Equilibrium,” in Russian, Nauka, Moscow, 1973. [6] F. H. Clarke, “Optimization and Nomsmooth Analysis,” John Wiley, New York, 1983. [7] I. Ekeland and R. Teman, “Analyse Convexe et Problems Variationelles,” Dunod and Gauthier Villars, Paris, 1972. [8] A. D. Ioffe and V. M. Tikhomirov, “Theory of Extremal Problems,” in Russian, Nauka, Moscow, 1974. [9] B. S. Mordukhovich, “Approximation Methods in Problems of Optimization and Control,” in Russian, Nauka, Moscow, 1988. [10] B. N. Pshenichnyi, “Convex Analysis and Extremal Problems,” in Russian, Nauka, Moscow, 1980. [11] R. P. Agarwal and D, O’Regan, “Fixed-Point Theory for Weakly Sequentially Upper-Semicontinuous Maps with Applications to Differential Inclusions,” Nonlinear Osccilations, Vol. 5, No. 3, 2002, pp. 277-286. doi:10.1023/A:1022312305912 [12] R. Vinter, “Optimal Control,” Birkh?user, Boston, 2000. [13] K. Wilfred, “Maxima and Minima with Applications, Practical Optimization and Duality,” John Wiley and Sons, Inc., New York, 1999. [14] W.-X. Zhong, “Duality System in Applied Mechanics and Optimal Control,” Springer, New York, 1992. [15] E. N. Mahmudov, “Approximation and Optimization of Discrete and Differential Inclusions,” Elsevier, New York, 2011. [16] E. N. Mahmudov, “Duality in the Problems of Optimal Control Described by First Order Partial Differential Inclusions,” Optimization: A Journal of Mathematical Programming and Operations Research, Vol. 59, No. 4, 2010, pp. 589-599. doi:10.1080/02331930802434666 [17] E. N. Mahmudov, “Duality in Optimal Control Problems of Optimal Control Described by Convex Discrete and Differential Inclusions with Delay,” Automat Remote Control, Vol. 48, No. 2, 1987, pp. 13-15. [18] E. N. Mahmudov, “Necessary and Sufficient Conditions for Discrete and Differential Inclusions of Elliptic Type,” Journal of Mathematical Analysis and Applications, Vol. 323, No. 2, 2006, pp. 768-789. doi:10.1016/j.jmaa.2005.10.069 [19] E. N. Mahmudov, “Locally Adjoint Mappings and Optimization of the First Boundary Value Problems for Hyperbolic Type Discrete and Differential Inclusions,” Non- linear Analysis: Theory, Methods & Applications, Vol. 67, No. 10, 2007, pp. 2966-2981. doi:10.1016/j.na.2006.09.054 [20] E. N. Mahmudov, “On Duality in Problems of Theory of Convex Difference Inclusions with Aftereffect,” (Russian) Differentsialnye Uravneniya, Vol. 146, No. 8, 1987, pp. 1315- 1324. [21] V. P. Mikhailov, “Partial Differantial Equations,” in Russian, Nauka, Moscow, 1976.
[1] A. Kurzhanski, “Set-Valued Analysis and Differential Inclusions,” Birkh?user, Boston, 1993. [2] J.-L. Lions, “Optimal Control of Systems Governed by Partial Differential Equations,” Springer, Berlin, 1991. [3] J. P. Aubin and A. Cellina, “Differential Inclusion,” Sp- ringer-Verlag, Berlin, 1984. [4] R. T. Rockafellar, “Convex Analysis,” 2nd Edition, Princeton University, Hoboken, 1972. [5] V. L. Makarov and A. M. Rubinov, “The Mathematical Theory of Economic Dynamics and Equilibrium,” in Russian, Nauka, Moscow, 1973. [6] F. H. Clarke, “Optimization and Nomsmooth Analysis,” John Wiley, New York, 1983. [7] I. Ekeland and R. Teman, “Analyse Convexe et Problems Variationelles,” Dunod and Gauthier Villars, Paris, 1972. [8] A. D. Ioffe and V. M. Tikhomirov, “Theory of Extremal Problems,” in Russian, Nauka, Moscow, 1974. [9] B. S. Mordukhovich, “Approximation Methods in Problems of Optimization and Control,” in Russian, Nauka, Moscow, 1988. [10] B. N. Pshenichnyi, “Convex Analysis and Extremal Problems,” in Russian, Nauka, Moscow, 1980. [11] R. P. Agarwal and D, O’Regan, “Fixed-Point Theory for Weakly Sequentially Upper-Semicontinuous Maps with Applications to Differential Inclusions,” Nonlinear Osccilations, Vol. 5, No. 3, 2002, pp. 277-286. doi:10.1023/A:1022312305912 [12] R. Vinter, “Optimal Control,” Birkh?user, Boston, 2000. [13] K. Wilfred, “Maxima and Minima with Applications, Practical Optimization and Duality,” John Wiley and Sons, Inc., New York, 1999. [14] W.-X. Zhong, “Duality System in Applied Mechanics and Optimal Control,” Springer, New York, 1992. [15] E. N. Mahmudov, “Approximation and Optimization of Discrete and Differential Inclusions,” Elsevier, New York, 2011. [16] E. N. Mahmudov, “Duality in the Problems of Optimal Control Described by First Order Partial Differential Inclusions,” Optimization: A Journal of Mathematical Programming and Operations Research, Vol. 59, No. 4, 2010, pp. 589-599. doi:10.1080/02331930802434666 [17] E. N. Mahmudov, “Duality in Optimal Control Problems of Optimal Control Described by Convex Discrete and Differential Inclusions with Delay,” Automat Remote Control, Vol. 48, No. 2, 1987, pp. 13-15. [18] E. N. Mahmudov, “Necessary and Sufficient Conditions for Discrete and Differential Inclusions of Elliptic Type,” Journal of Mathematical Analysis and Applications, Vol. 323, No. 2, 2006, pp. 768-789. doi:10.1016/j.jmaa.2005.10.069 [19] E. N. Mahmudov, “Locally Adjoint Mappings and Optimization of the First Boundary Value Problems for Hyperbolic Type Discrete and Differential Inclusions,” Non- linear Analysis: Theory, Methods & Applications, Vol. 67, No. 10, 2007, pp. 2966-2981. doi:10.1016/j.na.2006.09.054 [20] E. N. Mahmudov, “On Duality in Problems of Theory of Convex Difference Inclusions with Aftereffect,” (Russian) Differentsialnye Uravneniya, Vol. 146, No. 8, 1987, pp. 1315- 1324. [21] V. P. Mikhailov, “Partial Differantial Equations,” in Russian, Nauka, Moscow, 1976.