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Strict feasibility of variational inclusion problems in reflexive Banach spaces
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 |
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.
References:
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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. |
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N. U. Ahmed,
Differential inclusions operator valued measures and optimal control, Dynamic Syst. Appl., 16 (2007), 13-35.
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D. Azzam-Laouir and L. Sabrina,
Existence solutions for a class of second order differential inclusions, Pacific Journ. of Optim., 6 (2005), 339-346.
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A. Bagirov, N. Karmitsa and M. Makela, Introduction to Nonsmooth Optimization, Springer, 2014.
doi: 10.1007/978-3-319-08114-4. |
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A. Cernea,
Continuous version of Filippov's theorem for a Sturm-Liouville type differential inclusion, E.J. Differ. Equat., 2008 (2008), 1-7.
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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. |
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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. |
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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. |
| [9] |
Q. Liqun, K. Lay Teo and X. Yang, Optimization and Control with Applications, Springer, 2005.
doi: 10.1007/b104943. |
| [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. |
| [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. 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. |
| [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. 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. |
| [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. |
| [9] |
Q. Liqun, K. Lay Teo and X. Yang, Optimization and Control with Applications, Springer, 2005.
doi: 10.1007/b104943. |
| [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. |
| [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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