Sufficient Conditions of Optimality for Second Order Differential Inclusions

Gulgun Kayakutlu; Elimhan N. Mahmudov

doi:doi:10.11648/j.engmath.20170101.11

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Sufficient Conditions of Optimality for Second Order Differential Inclusions

Gulgun Kayakutlu, Elimhan N. Mahmudov

Published in Engineering Mathematics (Volume 1, Issue 1)

Received: 12 December 2016 Accepted: 21 December 2016 Published: 16 January 2017

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Abstract

In this paper we are concerned with the Bolza problem (P_C) for second order differential inclusions (SODIs). The aim is to derive sufficient conditions of optimality for a problem (P_C). The basic concept of obtaining these conditions is the locally adjoint mappings (LAMs). Besides the transversality conditions, approaches to the general problem therefore involve distinctive Euler-Lagrange and Hamiltonian kind of adjoint inclusions. Furthermore, the aim of the considered “linear” problem with SODIs is to show the reader, by example, how the obtained results can be applied in practice.

Published in Engineering Mathematics (Volume 1, Issue 1)

DOI 10.11648/j.engmath.20170101.11

Page(s) 1-6

Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

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Copyright © The Author(s), 2017. Published by Science Publishing Group

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Keywords

Differential Inclusion, Cauchy, Euler-Lagrange, Adjoint, Multivalued, Second Order, Transversality

References

[1] J. P. Aubin, H. Frankowska, Set-Valued Analysis, Systems Control Found. Appl., Birkhauser Boston, Inc., Boston, MA, 1990.

[2] A. Auslender, J. Mechler, “Second order viability problems for differential inclusions”, J. Math. Anal. Appl., Vol. 181, 1994, pp. 205-218.

[3] A. Boucherif and B. Chanane, “Boundary value problems for second order differential inclusions”, Int. J. Differ. Equ. Appl., Vol. 7, 2003, pp. 147–151.

[4] M. Benchohra and A. Ouahab, “Initial boundary value problems for second order impulsive functional differential inclusions”, E. J. Qualitative Theory of Diff. Equ., Vol. 3, 2003, pp. 1-10.

[5] A. Cernea, “On the existence of viable solutions for a class of second order differential inclusions”, Discuss. Math., Differ. Incl., Vol. 22, No. 1, 2002, pp. 6778.

[6] N. A. Izobov, Lyapunov Exponents and Stability (Stability Oscillations and Optimization of Systems), Cambridge Scientific Publishers, 2013.

[7] A. B. Kurzhanski, V. M. Veliov, Set-Valued Analysis and Differential Inclusions (Progress in Systems and Control Theory) Birkhäuser, 1993.

[8] A. V. Lotov, “Multicriteria optimization of convex dynamical systems”, Differential Equations Vol. 45, No 11, 2009, pp 1669-1680.

[9] V. Lupulescu, “A viability result for nonconvex second order differential inclusions”, Electronic J. Diff. Equ., Vol. 76, 2002, pp. 112.

[10] E. N. Mahmudov, “Optimization of Discrete inclusions with Distrubuted Parameters”, Optimization Vol. 21, No. 2, 1990, pp. 197-207.

[11] E. N. Mahmudov, “Sufficient conditions for optimality for differential inclusions of parabolic type and duality”, J. Global Optim., Vol. 41, No. 1, 2008, pp. 31-42.

[12] E. N. Mahmudov, “Optimization of Second Order Discrete Approximation Inclusions”, Num. Funct. Anal. Optim., DOI: 10.1080/ 01630563. 2015. 1014048

[13] E. N. Mahmudov, “Locally adjoint mappings and optimization of the first boundary value problem for hyperbolic type discrete and differential inclusions”, Nonlin. Anal., Vol. 67, No. 10, 2007, pp. 2966-2981.

[14] E. N. Mahmudov, “Necessary and sufficient conditions for discrete and differential inclusions of elliptic type”, J. Math. Anal. Appl. Vol. 323, No. 2, 2006, pp. 768-789.

[15] E. N. Mahmudov, Approximation and Optimization of Discrete and Differential Inclusions, Elsevier, 2011.

[16] L. Marco and J. A. Murillo, “Lyapunov functions for second order differential inclusions: A viability approach”, J. Math. Anal. Appl. Vol. 262, No. 1, 2001, pp. 39-354.

[17] B. S. Mordukhovich, “Variational Analysis and Generalized Differentiation, I: Basic Theory; II: Applications”, Grundlehren Series (Fundamental Principles of Mathematical Sciences), Vol. 330 and 331, Springer, 2006.

[18] N. P. Osmolovskii, “On Second-Order Necessary Conditions for Broken Extremals”, J. Optim. Theory Appl., Vol. 164, No. 2, 2015, pp. 379-406.

[19] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The Mathematical Theory of Optimal Processes. New York, London, Sydney, John Wiley & Sons, Inc., 1965.

[20] E. S. Polovinkin, G. V. Smirnov, “On a time-optimal control problem for differential inclusions”, J. Diff. Equ., Vol. 22, pp. 1986, 940-952.

[21] N. N. Rachkovskii, “Necessary Cone Optimality Conditions for Trajectories of Discrete Inclusions”, Diff. Equ., Vol. 39, No. 2, 2003, pp. 289–300.

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Gulgun Kayakutlu, Elimhan N. Mahmudov. (2017). Sufficient Conditions of Optimality for Second Order Differential Inclusions. Engineering Mathematics, 1(1), 1-6. https://doi.org/10.11648/j.engmath.20170101.11

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Gulgun Kayakutlu; Elimhan N. Mahmudov. Sufficient Conditions of Optimality for Second Order Differential Inclusions. Eng. Math. 2017, 1(1), 1-6. doi: 10.11648/j.engmath.20170101.11

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Gulgun Kayakutlu, Elimhan N. Mahmudov. Sufficient Conditions of Optimality for Second Order Differential Inclusions. Eng Math. 2017;1(1):1-6. doi: 10.11648/j.engmath.20170101.11

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@article{10.11648/j.engmath.20170101.11, author = {Gulgun Kayakutlu and Elimhan N. Mahmudov}, title = {Sufficient Conditions of Optimality for Second Order Differential Inclusions}, journal = {Engineering Mathematics}, volume = {1}, number = {1}, pages = {1-6}, doi = {10.11648/j.engmath.20170101.11}, url = {https://doi.org/10.11648/j.engmath.20170101.11}, eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.engmath.20170101.11}, abstract = {In this paper we are concerned with the Bolza problem (PC) for second order differential inclusions (SODIs). The aim is to derive sufficient conditions of optimality for a problem (PC). The basic concept of obtaining these conditions is the locally adjoint mappings (LAMs). Besides the transversality conditions, approaches to the general problem therefore involve distinctive Euler-Lagrange and Hamiltonian kind of adjoint inclusions. Furthermore, the aim of the considered “linear” problem with SODIs is to show the reader, by example, how the obtained results can be applied in practice.}, year = {2017} }

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TY - JOUR T1 - Sufficient Conditions of Optimality for Second Order Differential Inclusions AU - Gulgun Kayakutlu AU - Elimhan N. Mahmudov Y1 - 2017/01/16 PY - 2017 N1 - https://doi.org/10.11648/j.engmath.20170101.11 DO - 10.11648/j.engmath.20170101.11 T2 - Engineering Mathematics JF - Engineering Mathematics JO - Engineering Mathematics SP - 1 EP - 6 PB - Science Publishing Group SN - 2640-088X UR - https://doi.org/10.11648/j.engmath.20170101.11 AB - In this paper we are concerned with the Bolza problem (PC) for second order differential inclusions (SODIs). The aim is to derive sufficient conditions of optimality for a problem (PC). The basic concept of obtaining these conditions is the locally adjoint mappings (LAMs). Besides the transversality conditions, approaches to the general problem therefore involve distinctive Euler-Lagrange and Hamiltonian kind of adjoint inclusions. Furthermore, the aim of the considered “linear” problem with SODIs is to show the reader, by example, how the obtained results can be applied in practice. VL - 1 IS - 1 ER -

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Gulgun Kayakutlu

Industrial Engineering Department, Istanbul Technical University, Istanbul, Turkey

Elimhan N. Mahmudov

Department of Mathematics, Istanbul Technical University, Istanbul, Turkey

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Gulgun Kayakutlu, Elimhan N. Mahmudov. (2017). Sufficient Conditions of Optimality for Second Order Differential Inclusions. Engineering Mathematics, 1(1), 1-6. https://doi.org/10.11648/j.engmath.20170101.11

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Gulgun Kayakutlu; Elimhan N. Mahmudov. Sufficient Conditions of Optimality for Second Order Differential Inclusions. Eng. Math. 2017, 1(1), 1-6. doi: 10.11648/j.engmath.20170101.11

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Gulgun Kayakutlu, Elimhan N. Mahmudov. Sufficient Conditions of Optimality for Second Order Differential Inclusions. Eng Math. 2017;1(1):1-6. doi: 10.11648/j.engmath.20170101.11

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@article{10.11648/j.engmath.20170101.11, author = {Gulgun Kayakutlu and Elimhan N. Mahmudov}, title = {Sufficient Conditions of Optimality for Second Order Differential Inclusions}, journal = {Engineering Mathematics}, volume = {1}, number = {1}, pages = {1-6}, doi = {10.11648/j.engmath.20170101.11}, url = {https://doi.org/10.11648/j.engmath.20170101.11}, eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.engmath.20170101.11}, abstract = {In this paper we are concerned with the Bolza problem (PC) for second order differential inclusions (SODIs). The aim is to derive sufficient conditions of optimality for a problem (PC). The basic concept of obtaining these conditions is the locally adjoint mappings (LAMs). Besides the transversality conditions, approaches to the general problem therefore involve distinctive Euler-Lagrange and Hamiltonian kind of adjoint inclusions. Furthermore, the aim of the considered “linear” problem with SODIs is to show the reader, by example, how the obtained results can be applied in practice.}, year = {2017} }

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TY - JOUR T1 - Sufficient Conditions of Optimality for Second Order Differential Inclusions AU - Gulgun Kayakutlu AU - Elimhan N. Mahmudov Y1 - 2017/01/16 PY - 2017 N1 - https://doi.org/10.11648/j.engmath.20170101.11 DO - 10.11648/j.engmath.20170101.11 T2 - Engineering Mathematics JF - Engineering Mathematics JO - Engineering Mathematics SP - 1 EP - 6 PB - Science Publishing Group SN - 2640-088X UR - https://doi.org/10.11648/j.engmath.20170101.11 AB - In this paper we are concerned with the Bolza problem (PC) for second order differential inclusions (SODIs). The aim is to derive sufficient conditions of optimality for a problem (PC). The basic concept of obtaining these conditions is the locally adjoint mappings (LAMs). Besides the transversality conditions, approaches to the general problem therefore involve distinctive Euler-Lagrange and Hamiltonian kind of adjoint inclusions. Furthermore, the aim of the considered “linear” problem with SODIs is to show the reader, by example, how the obtained results can be applied in practice. VL - 1 IS - 1 ER -

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Published in	Engineering Mathematics (Volume 1, Issue 1)
DOI	10.11648/j.engmath.20170101.11
Page(s)	1-6
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2017. Published by Science Publishing Group

[1]	J. P. Aubin, H. Frankowska, Set-Valued Analysis, Systems Control Found. Appl., Birkhauser Boston, Inc., Boston, MA, 1990.
[2]	A. Auslender, J. Mechler, “Second order viability problems for differential inclusions”, J. Math. Anal. Appl., Vol. 181, 1994, pp. 205-218.
[3]	A. Boucherif and B. Chanane, “Boundary value problems for second order differential inclusions”, Int. J. Differ. Equ. Appl., Vol. 7, 2003, pp. 147–151.
[4]	M. Benchohra and A. Ouahab, “Initial boundary value problems for second order impulsive functional differential inclusions”, E. J. Qualitative Theory of Diff. Equ., Vol. 3, 2003, pp. 1-10.
[5]	A. Cernea, “On the existence of viable solutions for a class of second order differential inclusions”, Discuss. Math., Differ. Incl., Vol. 22, No. 1, 2002, pp. 6778.
[6]	N. A. Izobov, Lyapunov Exponents and Stability (Stability Oscillations and Optimization of Systems), Cambridge Scientific Publishers, 2013.
[7]	A. B. Kurzhanski, V. M. Veliov, Set-Valued Analysis and Differential Inclusions (Progress in Systems and Control Theory) Birkhäuser, 1993.
[8]	A. V. Lotov, “Multicriteria optimization of convex dynamical systems”, Differential Equations Vol. 45, No 11, 2009, pp 1669-1680.
[9]	V. Lupulescu, “A viability result for nonconvex second order differential inclusions”, Electronic J. Diff. Equ., Vol. 76, 2002, pp. 112.
[10]	E. N. Mahmudov, “Optimization of Discrete inclusions with Distrubuted Parameters”, Optimization Vol. 21, No. 2, 1990, pp. 197-207.
[11]	E. N. Mahmudov, “Sufficient conditions for optimality for differential inclusions of parabolic type and duality”, J. Global Optim., Vol. 41, No. 1, 2008, pp. 31-42.
[12]	E. N. Mahmudov, “Optimization of Second Order Discrete Approximation Inclusions”, Num. Funct. Anal. Optim., DOI: 10.1080/ 01630563. 2015. 1014048
[13]	E. N. Mahmudov, “Locally adjoint mappings and optimization of the first boundary value problem for hyperbolic type discrete and differential inclusions”, Nonlin. Anal., Vol. 67, No. 10, 2007, pp. 2966-2981.
[14]	E. N. Mahmudov, “Necessary and sufficient conditions for discrete and differential inclusions of elliptic type”, J. Math. Anal. Appl. Vol. 323, No. 2, 2006, pp. 768-789.
[15]	E. N. Mahmudov, Approximation and Optimization of Discrete and Differential Inclusions, Elsevier, 2011.
[16]	L. Marco and J. A. Murillo, “Lyapunov functions for second order differential inclusions: A viability approach”, J. Math. Anal. Appl. Vol. 262, No. 1, 2001, pp. 39-354.
[17]	B. S. Mordukhovich, “Variational Analysis and Generalized Differentiation, I: Basic Theory; II: Applications”, Grundlehren Series (Fundamental Principles of Mathematical Sciences), Vol. 330 and 331, Springer, 2006.
[18]	N. P. Osmolovskii, “On Second-Order Necessary Conditions for Broken Extremals”, J. Optim. Theory Appl., Vol. 164, No. 2, 2015, pp. 379-406.
[19]	L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The Mathematical Theory of Optimal Processes. New York, London, Sydney, John Wiley & Sons, Inc., 1965.
[20]	E. S. Polovinkin, G. V. Smirnov, “On a time-optimal control problem for differential inclusions”, J. Diff. Equ., Vol. 22, pp. 1986, 940-952.
[21]	N. N. Rachkovskii, “Necessary Cone Optimality Conditions for Trajectories of Discrete Inclusions”, Diff. Equ., Vol. 39, No. 2, 2003, pp. 289–300.