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March  2020, 10(1): 93-106. doi: 10.3934/naco.2019035

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

Savin Treanţă

University Politehnica of Bucharest, Faculty of Applied Sciences, Department of Applied Mathematics, 313 Splaiul Independentei, 060042 Bucharest, Romania

Received  October 2018 Revised  March 2019 Published  May 2019

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.

Keywords: Multidimensional vector control problem, Kuhn-Tucker optimality conditions, V-KT-pseudoinvexity, multiple integral cost functional, flow.
Mathematics Subject Classification: Primary: 26B25, 65K10, 90C29; Secondary: 90C30, 49K20, 46T20, 58J32.
Citation: Savin Treanţă. Characterization of efficient solutions for a class of PDE-constrained vector control problems. Numerical Algebra, Control & Optimization, 2020, 10 (1) : 93-106. doi: 10.3934/naco.2019035
References:
[1]

V. M. Alekseev, M. V. Tikhomirov and S. V. Fomin, Commande Optimale, Mir, Moscow, 1982.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

D. H. Martin, The essence of invexity, J. Optim. Theory Appl., 47 (1985), 65-76.  doi: 10.1007/BF00941316.  Google Scholar

[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.  Google Scholar

[12]

B. Mond and M. A. Hanson, Duality for control problems, SIAM J. Control, 6 (1968), 114-120.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[17]

V. Prepeliţă, Stability of a class of multidimensional continuous-discrete linear systems, Math. Reports, 9 (2007), 387-398.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

show all references

References:
[1]

V. M. Alekseev, M. V. Tikhomirov and S. V. Fomin, Commande Optimale, Mir, Moscow, 1982.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

D. H. Martin, The essence of invexity, J. Optim. Theory Appl., 47 (1985), 65-76.  doi: 10.1007/BF00941316.  Google Scholar

[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.  Google Scholar

[12]

B. Mond and M. A. Hanson, Duality for control problems, SIAM J. Control, 6 (1968), 114-120.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[17]

V. Prepeliţă, Stability of a class of multidimensional continuous-discrete linear systems, Math. Reports, 9 (2007), 387-398.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

Figure 1.  Graphical illustrations for x(t) and u(t)
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