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

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

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.

Citation: Elimhan N. Mahmudov. Optimal control of Sturm-Liouville type evolution differential inclusions with endpoint constraints. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2503-2520. doi: 10.3934/jimo.2019066
References:
[1]

S. AdlyA. 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.

[2]

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

[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. 

[4]

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

[5]

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

[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.

[7]

Y. GaoX. YangJ. 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.

[8]

S. J. LiS. 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.

[9]

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

[10]

Y. LiuJ. 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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

show all references

References:
[1]

S. AdlyA. 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.

[2]

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

[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. 

[4]

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

[5]

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

[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.

[7]

Y. GaoX. YangJ. 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.

[8]

S. J. LiS. 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.

[9]

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

[10]

Y. LiuJ. 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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

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