American Institute of Mathematical Sciences

Advanced
September  2020, 16(5): 2503-2520. doi: 10.3934/jimo.2019066

Optimal control of Sturm-Liouville type evolution differential inclusions with endpoint constraints

Elimhan N. Mahmudov 1,2,,

1. 

Department of Mathematics, Istanbul Technical University, 34469, Maslak, Istanbul, Turkey
2. 

Azerbaijan National Academy of Sciences Institute of Control Systems, Baku, Azerbaijan

* Corresponding author: Elimhan N. Mahmudov

Received  November 2018 Revised  February 2019 Published  September 2020 Early access  May 2019

The present paper studies a new class of problems of optimal control theory with linear second order self-adjoint Sturm-Liouville type differential operators and with functional and non-functional endpoint constraints. Sufficient conditions of optimality, containing both the second order Euler-Lagrange and Hamiltonian type inclusions are derived. The presence of functional constraints generates a special second order transversality inclusions and complementary slackness conditions peculiar to inequality constraints; this approach and results make a bridge between optimal control problem with Sturm-Liouville type differential differential inclusions and constrained mathematical programming problems in finite-dimensional spaces.The idea for obtaining optimality conditions is based on applying locally-adjoint mappings to Sturm-Liouville type set-valued mappings. The result generalizes to the problem with a second order non-self-adjoint differential operator. Furthermore, practical applications of these results are demonstrated by optimization of some semilinear optimal control problems for which the Pontryagin maximum condition is obtained. A numerical example is given to illustrate the feasibility and effectiveness of the theoretic results obtained.

Keywords: Sturm-Liouville, self-adjoint, functional, transversality, complementary slackness, Hamiltonian, set-valued.
Mathematics Subject Classification: Primary: 34A60, 34G25; Secondary: 65K05, 90C25.
Citation: Elimhan N. Mahmudov. Optimal control of Sturm-Liouville type evolution differential inclusions with endpoint constraints. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2503-2520. doi: 10.3934/jimo.2019066
References:
[1]

S. Adly, A. Hantoute and M. Th'era, Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain, Mathem. Program., 157 (2016), 349-374.  doi: 10.1007/s10107-015-0938-6.  Google Scholar

[2]

N. U. Ahmed, Differential inclusions operator valued measures and optimal control, Dynamic Syst. Appl., 16 (2007), 13-35.   Google Scholar

[3]

D. Azzam-Laouir and L. Sabrina, Existence solutions for a class of second order differential inclusions, Pacific Journ. of Optim., 6 (2005), 339-346.   Google Scholar

[4]

A. Bagirov, N. Karmitsa and M. Makela, Introduction to Nonsmooth Optimization, Springer, 2014. doi: 10.1007/978-3-319-08114-4.  Google Scholar

[5]

A. Cernea, Continuous version of Filippov's theorem for a Sturm-Liouville type differential inclusion, E.J. Differ. Equat., 2008 (2008), 1-7.   Google Scholar

[6]

F. H. Clarke, Functional Analysis, Calculus of Variations and Optimal Control, Graduate Texts in Mathematics, 264, Springer, 2013. doi: 10.1007/978-1-4471-4820-3.  Google Scholar

[7]

Y. Gao, X. Yang, J. Yang and H. Yan, Scalarizations and Lagrange multipliers for approximat solutions in the vector optimization problems with set-valued maps, J. Industrial Manag. Optim., 11 (2014), 673-683.  doi: 10.3934/jimo.2015.11.673.  Google Scholar

[8]

S. J. Li, S. K. Zhu and K. Lay Teo, New generalized second-order contingent epiderivatives and set-valued optimization problems, J. Optim. Theory Appl., 152 (2012), 587-604.  doi: 10.1007/s10957-011-9915-2.  Google Scholar

[9]

Q. Liqun, K. Lay Teo and X. Yang, Optimization and Control with Applications, Springer, 2005. doi: 10.1007/b104943.  Google Scholar

[10]

Y. Liu, J. Wu and Z. Li, Impulsive boundary value problems for Sturm-Liouville type differential inclusions, J. Syst. Sci. Complexity, 20 (2007), 370-380.  doi: 10.1007/s11424-007-9032-3.  Google Scholar

[11]

P. D. Loewen and R. T. Rockafellar, Optimal control of unbounded differential inclusions, SIAM J Contr Optim., 32 (1994), 442-470.  doi: 10.1137/S0363012991217494.  Google Scholar

[12]

E. N. Mahmudov, Approximation and Optimization of Discrete and Differential Inclusions, Elsevier: Boston, USA, 2011. doi: 10.1016/B978-0-12-388428-2.00001-1.  Google Scholar

[13]

E. N. Mahmudov, Approximation and Optimization of Higher order discrete and differential inclusions, Nonlin. Diff. Equat. Appl. (NoDEA), 21 (2014), 1-26.  doi: 10.1007/s00030-013-0234-1.  Google Scholar

[14]

E. N. Mahmudov, Optimal control of second order delay-discrete and delay differential inclusions with state constraints, Evol. Equat. Cont. Theory (EECT), 7 (2018), 501-529.  doi: 10.3934/eect.2018024.  Google Scholar

[15]

E. N. Mahmudov, Optimization of Fourth-Order Differential Inclusions, Proceed. Institute Mathem. Mechanics, 44 (2018), 90-106.   Google Scholar

[16]

E. N. Mahmudov, Optimization of second-order discrete approximation inclusions, Numeric. Funct. Anal. Optim., 36 (2015), 624-643.  doi: 10.1080/01630563.2015.1014048.  Google Scholar

[17]

E. N. Mahmudov, Optimization of Mayer problem with Sturm-Liouville-type differential inclusions, J. Optim, Theory Appl., 177 (2018), 345-375.  doi: 10.1007/s10957-018-1260-2.  Google Scholar

[18]

E. N. Mahmudov, Optimization of fourth order Sturm-Liouville type differential inclusions with initial point constraints, J. Industrial Manag. Optim., (2018).  doi: 10.3934/jimo.2018145.  Google Scholar

[19]

B. S. Mordukhovich, Optimal control of semilinear unbounded evolution inclusions with functional constraints, J. Optim. Theory Appl., 167 (2015), 821-841.  doi: 10.1007/s10957-013-0301-0.  Google Scholar

[20]

Y. Xu and Z. Peng, Higher-order sensitivity analysis in set-valued optimization under Henig efficiency, J. Industrial Manag. Optim., 13 (2017), 313-327.  doi: 10.3934/jimo.2016019.  Google Scholar

show all references

References:
[1]

S. Adly, A. Hantoute and M. Th'era, Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain, Mathem. Program., 157 (2016), 349-374.  doi: 10.1007/s10107-015-0938-6.  Google Scholar

[2]

N. U. Ahmed, Differential inclusions operator valued measures and optimal control, Dynamic Syst. Appl., 16 (2007), 13-35.   Google Scholar

[3]

D. Azzam-Laouir and L. Sabrina, Existence solutions for a class of second order differential inclusions, Pacific Journ. of Optim., 6 (2005), 339-346.   Google Scholar

[4]

A. Bagirov, N. Karmitsa and M. Makela, Introduction to Nonsmooth Optimization, Springer, 2014. doi: 10.1007/978-3-319-08114-4.  Google Scholar

[5]

A. Cernea, Continuous version of Filippov's theorem for a Sturm-Liouville type differential inclusion, E.J. Differ. Equat., 2008 (2008), 1-7.   Google Scholar

[6]

F. H. Clarke, Functional Analysis, Calculus of Variations and Optimal Control, Graduate Texts in Mathematics, 264, Springer, 2013. doi: 10.1007/978-1-4471-4820-3.  Google Scholar

[7]

Y. Gao, X. Yang, J. Yang and H. Yan, Scalarizations and Lagrange multipliers for approximat solutions in the vector optimization problems with set-valued maps, J. Industrial Manag. Optim., 11 (2014), 673-683.  doi: 10.3934/jimo.2015.11.673.  Google Scholar

[8]

S. J. Li, S. K. Zhu and K. Lay Teo, New generalized second-order contingent epiderivatives and set-valued optimization problems, J. Optim. Theory Appl., 152 (2012), 587-604.  doi: 10.1007/s10957-011-9915-2.  Google Scholar

[9]

Q. Liqun, K. Lay Teo and X. Yang, Optimization and Control with Applications, Springer, 2005. doi: 10.1007/b104943.  Google Scholar

[10]

Y. Liu, J. Wu and Z. Li, Impulsive boundary value problems for Sturm-Liouville type differential inclusions, J. Syst. Sci. Complexity, 20 (2007), 370-380.  doi: 10.1007/s11424-007-9032-3.  Google Scholar

[11]

P. D. Loewen and R. T. Rockafellar, Optimal control of unbounded differential inclusions, SIAM J Contr Optim., 32 (1994), 442-470.  doi: 10.1137/S0363012991217494.  Google Scholar

[12]

E. N. Mahmudov, Approximation and Optimization of Discrete and Differential Inclusions, Elsevier: Boston, USA, 2011. doi: 10.1016/B978-0-12-388428-2.00001-1.  Google Scholar

[13]

E. N. Mahmudov, Approximation and Optimization of Higher order discrete and differential inclusions, Nonlin. Diff. Equat. Appl. (NoDEA), 21 (2014), 1-26.  doi: 10.1007/s00030-013-0234-1.  Google Scholar

[14]

E. N. Mahmudov, Optimal control of second order delay-discrete and delay differential inclusions with state constraints, Evol. Equat. Cont. Theory (EECT), 7 (2018), 501-529.  doi: 10.3934/eect.2018024.  Google Scholar

[15]

E. N. Mahmudov, Optimization of Fourth-Order Differential Inclusions, Proceed. Institute Mathem. Mechanics, 44 (2018), 90-106.   Google Scholar

[16]

E. N. Mahmudov, Optimization of second-order discrete approximation inclusions, Numeric. Funct. Anal. Optim., 36 (2015), 624-643.  doi: 10.1080/01630563.2015.1014048.  Google Scholar

[17]

E. N. Mahmudov, Optimization of Mayer problem with Sturm-Liouville-type differential inclusions, J. Optim, Theory Appl., 177 (2018), 345-375.  doi: 10.1007/s10957-018-1260-2.  Google Scholar

[18]

E. N. Mahmudov, Optimization of fourth order Sturm-Liouville type differential inclusions with initial point constraints, J. Industrial Manag. Optim., (2018).  doi: 10.3934/jimo.2018145.  Google Scholar

[19]

B. S. Mordukhovich, Optimal control of semilinear unbounded evolution inclusions with functional constraints, J. Optim. Theory Appl., 167 (2015), 821-841.  doi: 10.1007/s10957-013-0301-0.  Google Scholar

[20]

Y. Xu and Z. Peng, Higher-order sensitivity analysis in set-valued optimization under Henig efficiency, J. Industrial Manag. Optim., 13 (2017), 313-327.  doi: 10.3934/jimo.2016019.  Google Scholar

[1]

Nuno Costa Dias, Andrea Posilicano, João Nuno Prata. Self-adjoint, globally defined Hamiltonian operators for systems with boundaries. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1687-1706. doi: 10.3934/cpaa.2011.10.1687
[2]

Russell Johnson, Luca Zampogni. On the inverse Sturm-Liouville problem. Discrete & Continuous Dynamical Systems, 2007, 18 (2&3) : 405-428. doi: 10.3934/dcds.2007.18.405
[3]

N. A. Chernyavskaya, L. A. Shuster. Spaces admissible for the Sturm-Liouville equation. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1023-1052. doi: 10.3934/cpaa.2018050
[4]

Abdallah El Hamidi, Aziz Hamdouni, Marwan Saleh. On eigenelements sensitivity for compact self-adjoint operators and applications. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 445-455. doi: 10.3934/dcdss.2016006
[5]

Peter Howard, Alim Sukhtayev. The Maslov and Morse indices for Sturm-Liouville systems on the half-line. Discrete & Continuous Dynamical Systems, 2020, 40 (2) : 983-1012. doi: 10.3934/dcds.2020068
[6]

Chuan-Fu Yang, Natalia Pavlovna Bondarenko. A partial inverse problem for the Sturm-Liouville operator on the lasso-graph. Inverse Problems & Imaging, 2019, 13 (1) : 69-79. doi: 10.3934/ipi.2019004
[7]

Roger Metzger, Carlos Arnoldo Morales Rojas, Phillipe Thieullen. Topological stability in set-valued dynamics. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1965-1975. doi: 10.3934/dcdsb.2017115
[8]

Geng-Hua Li, Sheng-Jie Li. Unified optimality conditions for set-valued optimizations. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1101-1116. doi: 10.3934/jimo.2018087
[9]

Dante Carrasco-Olivera, Roger Metzger Alvan, Carlos Arnoldo Morales Rojas. Topological entropy for set-valued maps. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3461-3474. doi: 10.3934/dcdsb.2015.20.3461
[10]

Yu Zhang, Tao Chen. Minimax problems for set-valued mappings with set optimization. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 327-340. doi: 10.3934/naco.2014.4.327
[11]

Guglielmo Feltrin. Multiple positive solutions of a sturm-liouville boundary value problem with conflicting nonlinearities. Communications on Pure & Applied Analysis, 2017, 16 (3) : 1083-1102. doi: 10.3934/cpaa.2017052
[12]

Rashad M. Asharabi, Jürgen Prestin. Computing eigenpairs of two-parameter Sturm-Liouville systems using the bivariate sinc-Gauss formula. Communications on Pure & Applied Analysis, 2020, 19 (8) : 4143-4158. doi: 10.3934/cpaa.2020185
[13]

Chuan-Fu Yang, Natalia Pavlovna Bondarenko, Xiao-Chuan Xu. An inverse problem for the Sturm-Liouville pencil with arbitrary entire functions in the boundary condition. Inverse Problems & Imaging, 2020, 14 (1) : 153-169. doi: 10.3934/ipi.2019068
[14]

Elimhan N. Mahmudov. Optimization of fourth order Sturm-Liouville type differential inclusions with initial point constraints. Journal of Industrial & Management Optimization, 2020, 16 (1) : 169-187. doi: 10.3934/jimo.2018145
[15]

Raziye Mert, Thabet Abdeljawad, Allan Peterson. A Sturm-Liouville approach for continuous and discrete Mittag-Leffler kernel fractional operators. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2417-2434. doi: 10.3934/dcdss.2020171
[16]

Qingbang Zhang, Caozong Cheng, Xuanxuan Li. Generalized minimax theorems for two set-valued mappings. Journal of Industrial & Management Optimization, 2013, 9 (1) : 1-12. doi: 10.3934/jimo.2013.9.1
[17]

Sina Greenwood, Rolf Suabedissen. 2-manifolds and inverse limits of set-valued functions on intervals. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5693-5706. doi: 10.3934/dcds.2017246
[18]

Zhenhua Peng, Zhongping Wan, Weizhi Xiong. Sensitivity analysis in set-valued optimization under strictly minimal efficiency. Evolution Equations & Control Theory, 2017, 6 (3) : 427-436. doi: 10.3934/eect.2017022
[19]

Mariusz Michta. Stochastic inclusions with non-continuous set-valued operators. Conference Publications, 2009, 2009 (Special) : 548-557. doi: 10.3934/proc.2009.2009.548
[20]

Guolin Yu. Topological properties of Henig globally efficient solutions of set-valued problems. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 309-316. doi: 10.3934/naco.2014.4.309

2020 Impact Factor: 1.801

Tools

Metrics

  • PDF downloads (128)
  • HTML views (611)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences