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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

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