Advanced Search
Article Contents
Article Contents

Characterization of efficient solutions for a class of PDE-constrained vector control problems

Abstract Full Text(HTML) Figure(1) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: Primary: 26B25, 65K10, 90C29; Secondary: 90C30, 49K20, 46T20, 58J32.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Graphical illustrations for x(t) and u(t)

  • [1] V. M. Alekseev, M. V. Tikhomirov and S. V. Fomin, Commande Optimale, Mir, Moscow, 1982.
    [2] M. Arana-JiménezR. Osuna-GómezA. 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. DiracV. A. Fock and B. Podolski, On quantum electrodynamics, Physikalische Zeitschrift der Sowjetunion, 2 (1932), 468-479. 
    [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.
    [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.
    [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. 
  • 加载中



Article Metrics

HTML views(1401) PDF downloads(289) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint