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Existence and iterative approximation method for solving mixed equilibrium problem under generalized monotonicity in Banach spaces
Characterization of efficient solutions for a class of PDE-constrained vector control problems
University Politehnica of Bucharest, Faculty of Applied Sciences, Department of Applied Mathematics, 313 Splaiul Independentei, 060042 Bucharest, Romania |
In this paper, we define a V-KT-pseudoinvex multidimensional vector control problem. More precisely, we introduce a new condition on the functionals which are involved in a multidimensional multiobjective (vector) control problem and we prove that a V-KT-pseudoinvex multidimensional vector control problem is characterized so that all Kuhn-Tucker points are efficient solutions. Also, the theoretical results derived in this paper are illustrated with an application.
References:
[1] |
V. M. Alekseev, M. V. Tikhomirov and S. V. Fomin, Commande Optimale, Mir, Moscow, 1982. |
[2] |
M. Arana-Jiménez, R. Osuna-Gómez, A. Rufián-Lizana and G. Ruiz-Garzón,
KT-invex control problem, Appl. Math. Comput., 197 (2008), 489-496.
doi: 10.1016/j.amc.2007.07.064. |
[3] |
F. Cardin and C. Viterbo,
Commuting Hamiltonians and Hamilton-Jacobi multi-time equations, Duke Math. J., 144 (2008), 235-284.
doi: 10.1215/00127094-2008-036. |
[4] |
D. A. Deckert and L. Nickel, Consistency of multi-time Dirac equations with general interaction potentials, J. Math. Phys., 57 (2016), 072301.
doi: 10.1063/1.4954947. |
[5] |
P. A. M. Dirac, V. A. Fock and B. Podolski, On quantum electrodynamics, Physikalische Zeitschrift der Sowjetunion, 2 (1932), 468-479. Google Scholar |
[6] |
A. Friedman,
The Cauchy problem in several time variables, Journal of Mathematics and Mechanics (Indiana Univ. Math. J.), 11 (1962), 859-889.
|
[7] |
S. Keppeler and M. Sieber, Particle creation and annihilation at interior boundaries: One-dimensional models, Preprint, arXiv: 1511.03071.
doi: 10.1088/1751-8113/49/12/125204. |
[8] |
W. S. Kendall,
Contours of Brownian processes with several-dimensional times, Probability Theory and Related Fields, 52 (1980), 267-276.
doi: 10.1007/BF00538891. |
[9] |
M. Lienert and L. Nickel, A simple explicitly solvable interacting relativistic $N$-particle model, J. Phys. A: Math. Theor., 48 (2015), 325301.
doi: 10.1088/1751-8113/48/32/325301. |
[10] |
D. H. Martin,
The essence of invexity, J. Optim. Theory Appl., 47 (1985), 65-76.
doi: 10.1007/BF00941316. |
[11] |
Şt. Mititelu and S. Treanţă,
Efficiency conditions in vector control problems governed by multiple integrals, J. Appl. Math. Comput., 57 (2018), 647-665.
doi: 10.1007/s12190-017-1126-z. |
[12] |
B. Mond and M. A. Hanson,
Duality for control problems, SIAM J. Control, 6 (1968), 114-120.
|
[13] |
B. Mond and I. Smart,
Duality and sufficiency in control problems with invexity, J. Math. Anal. Appl., 136 (1988), 325-333.
doi: 10.1016/0022-247X(88)90135-7. |
[14] |
M. Motta and F. Rampazzo,
Nonsmooth multi-time Hamilton-Jacobi systems, Indiana Univ. Math. J., 55 (2006), 1573-1614.
doi: 10.1512/iumj.2006.55.2760. |
[15] |
S. Petrat and R. Tumulka,
Multi-time wave functions for quantum field theory, Ann. Phys., 345 (2014), 17-54.
doi: 10.1016/j.aop.2014.03.004. |
[16] |
V. Preda,
On duality and sufficiency in control problems with general invexity, Bull. Math. de la Soc. Sci. Math de Roumanie, 35 (1991), 271-280.
|
[17] |
V. Prepeliţă,
Stability of a class of multidimensional continuous-discrete linear systems, Math. Reports, 9 (2007), 387-398.
|
[18] |
D. J. Saunders, The Geometry of Jet Bundles, London Math. Soc. Lecture Notes Series, 142 (1989), Cambridge Univ. Press, Cambridge
doi: 10.1017/CBO9780511526411. |
[19] |
S. Teufel and R. Tumulka, New type of Hamiltonians without ultraviolet divergence for quantum field theories, Preprint, https://arxiv.org/abs/1505.04847v1. Google Scholar |
[20] |
S. Tomonaga,
On a relativistically invariant formulation of the quantum theory of wave fields, Progress of Theoretical Physics, 1 (1946), 27-42.
doi: 10.1080/10724117.1994.11974884. |
[21] |
S. Treanţă,
PDEs of Hamilton-Pfaff type via multi-time optimization problems, U.P.B. Sci. Bull., Series A: Appl. Math. Phys., 76 (2014), 163-168.
|
[22] |
S. Treanţă, Optimal control problems on higher order jet bundles, The Intern. Conf. "Differential Geometry - Dynamical Systems", October 10-13, 2013, Bucharest-Romania, Balkan Society of Geometers, Geometry Balkan Press (2014), 181–192. |
[23] |
S. Treanţă,
Multiobjective fractional variational problem on higher-order jet bundles, Commun. Math. Stat., 4 (2016), 323-340.
doi: 10.1007/s40304-016-0087-0. |
[24] |
S. Treanţă,
Higher-order Hamilton dynamics and Hamilton-Jacobi divergence PDE, Comput. Math. Appl., 75 (2018), 547-560.
doi: 10.1016/j.camwa.2017.09.033. |
[25] |
S. Treanţă and M. Arana-Jiménez,
KT-pseudoinvex multidimensional control problem, Optim. Control Appl. Meth., 39 (2018), 1291-1300.
doi: 10.1002/oca.2410. |
[26] |
S. Treanţă and M. Arana-Jiménez,
On generalized KT-pseudoinvex control problems involving multiple integral functionals, Eur. J. Control, 43 (2018), 39-45.
doi: 10.1016/j.ejcon.2018.05.004. |
[27] |
S. Treanţă,
On a new class of vector variational control problems, Numer. Func. Anal. Opt., 39 (2018), 1594-1603.
doi: 10.1080/01630563.2018.1488142. |
[28] |
C. Udrişte and I. Ţevy,
Multitime dynamic programming for multiple integral actions, J. Glob. Optim., 51 (2011), 345-360.
doi: 10.1007/s10898-010-9599-4. |
[29] |
G-W. Weber, F. Yilmaz, H.Ö. Bakan and E. Savku, Approximation of Optimal Stochastic Control Problems for Multi-dimensional Stochastic Differential Equations by Using Itô-Taylor Method with Malliavin Calculus, The 9th International Conference on Optimization: Techniques and Applications, Taipei, Taiwan, 2013. Google Scholar |
[30] |
N. I. Yurchuk,
A partially characteristic mixed boundary value problem with Goursat initial conditions for linear equations with two-dimensional time, Diff. Uravn., 5 (1969), 898-910.
|
show all references
References:
[1] |
V. M. Alekseev, M. V. Tikhomirov and S. V. Fomin, Commande Optimale, Mir, Moscow, 1982. |
[2] |
M. Arana-Jiménez, R. Osuna-Gómez, A. Rufián-Lizana and G. Ruiz-Garzón,
KT-invex control problem, Appl. Math. Comput., 197 (2008), 489-496.
doi: 10.1016/j.amc.2007.07.064. |
[3] |
F. Cardin and C. Viterbo,
Commuting Hamiltonians and Hamilton-Jacobi multi-time equations, Duke Math. J., 144 (2008), 235-284.
doi: 10.1215/00127094-2008-036. |
[4] |
D. A. Deckert and L. Nickel, Consistency of multi-time Dirac equations with general interaction potentials, J. Math. Phys., 57 (2016), 072301.
doi: 10.1063/1.4954947. |
[5] |
P. A. M. Dirac, V. A. Fock and B. Podolski, On quantum electrodynamics, Physikalische Zeitschrift der Sowjetunion, 2 (1932), 468-479. Google Scholar |
[6] |
A. Friedman,
The Cauchy problem in several time variables, Journal of Mathematics and Mechanics (Indiana Univ. Math. J.), 11 (1962), 859-889.
|
[7] |
S. Keppeler and M. Sieber, Particle creation and annihilation at interior boundaries: One-dimensional models, Preprint, arXiv: 1511.03071.
doi: 10.1088/1751-8113/49/12/125204. |
[8] |
W. S. Kendall,
Contours of Brownian processes with several-dimensional times, Probability Theory and Related Fields, 52 (1980), 267-276.
doi: 10.1007/BF00538891. |
[9] |
M. Lienert and L. Nickel, A simple explicitly solvable interacting relativistic $N$-particle model, J. Phys. A: Math. Theor., 48 (2015), 325301.
doi: 10.1088/1751-8113/48/32/325301. |
[10] |
D. H. Martin,
The essence of invexity, J. Optim. Theory Appl., 47 (1985), 65-76.
doi: 10.1007/BF00941316. |
[11] |
Şt. Mititelu and S. Treanţă,
Efficiency conditions in vector control problems governed by multiple integrals, J. Appl. Math. Comput., 57 (2018), 647-665.
doi: 10.1007/s12190-017-1126-z. |
[12] |
B. Mond and M. A. Hanson,
Duality for control problems, SIAM J. Control, 6 (1968), 114-120.
|
[13] |
B. Mond and I. Smart,
Duality and sufficiency in control problems with invexity, J. Math. Anal. Appl., 136 (1988), 325-333.
doi: 10.1016/0022-247X(88)90135-7. |
[14] |
M. Motta and F. Rampazzo,
Nonsmooth multi-time Hamilton-Jacobi systems, Indiana Univ. Math. J., 55 (2006), 1573-1614.
doi: 10.1512/iumj.2006.55.2760. |
[15] |
S. Petrat and R. Tumulka,
Multi-time wave functions for quantum field theory, Ann. Phys., 345 (2014), 17-54.
doi: 10.1016/j.aop.2014.03.004. |
[16] |
V. Preda,
On duality and sufficiency in control problems with general invexity, Bull. Math. de la Soc. Sci. Math de Roumanie, 35 (1991), 271-280.
|
[17] |
V. Prepeliţă,
Stability of a class of multidimensional continuous-discrete linear systems, Math. Reports, 9 (2007), 387-398.
|
[18] |
D. J. Saunders, The Geometry of Jet Bundles, London Math. Soc. Lecture Notes Series, 142 (1989), Cambridge Univ. Press, Cambridge
doi: 10.1017/CBO9780511526411. |
[19] |
S. Teufel and R. Tumulka, New type of Hamiltonians without ultraviolet divergence for quantum field theories, Preprint, https://arxiv.org/abs/1505.04847v1. Google Scholar |
[20] |
S. Tomonaga,
On a relativistically invariant formulation of the quantum theory of wave fields, Progress of Theoretical Physics, 1 (1946), 27-42.
doi: 10.1080/10724117.1994.11974884. |
[21] |
S. Treanţă,
PDEs of Hamilton-Pfaff type via multi-time optimization problems, U.P.B. Sci. Bull., Series A: Appl. Math. Phys., 76 (2014), 163-168.
|
[22] |
S. Treanţă, Optimal control problems on higher order jet bundles, The Intern. Conf. "Differential Geometry - Dynamical Systems", October 10-13, 2013, Bucharest-Romania, Balkan Society of Geometers, Geometry Balkan Press (2014), 181–192. |
[23] |
S. Treanţă,
Multiobjective fractional variational problem on higher-order jet bundles, Commun. Math. Stat., 4 (2016), 323-340.
doi: 10.1007/s40304-016-0087-0. |
[24] |
S. Treanţă,
Higher-order Hamilton dynamics and Hamilton-Jacobi divergence PDE, Comput. Math. Appl., 75 (2018), 547-560.
doi: 10.1016/j.camwa.2017.09.033. |
[25] |
S. Treanţă and M. Arana-Jiménez,
KT-pseudoinvex multidimensional control problem, Optim. Control Appl. Meth., 39 (2018), 1291-1300.
doi: 10.1002/oca.2410. |
[26] |
S. Treanţă and M. Arana-Jiménez,
On generalized KT-pseudoinvex control problems involving multiple integral functionals, Eur. J. Control, 43 (2018), 39-45.
doi: 10.1016/j.ejcon.2018.05.004. |
[27] |
S. Treanţă,
On a new class of vector variational control problems, Numer. Func. Anal. Opt., 39 (2018), 1594-1603.
doi: 10.1080/01630563.2018.1488142. |
[28] |
C. Udrişte and I. Ţevy,
Multitime dynamic programming for multiple integral actions, J. Glob. Optim., 51 (2011), 345-360.
doi: 10.1007/s10898-010-9599-4. |
[29] |
G-W. Weber, F. Yilmaz, H.Ö. Bakan and E. Savku, Approximation of Optimal Stochastic Control Problems for Multi-dimensional Stochastic Differential Equations by Using Itô-Taylor Method with Malliavin Calculus, The 9th International Conference on Optimization: Techniques and Applications, Taipei, Taiwan, 2013. Google Scholar |
[30] |
N. I. Yurchuk,
A partially characteristic mixed boundary value problem with Goursat initial conditions for linear equations with two-dimensional time, Diff. Uravn., 5 (1969), 898-910.
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