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

  • * Corresponding author: Elimhan N. Mahmudov

    * Corresponding author: Elimhan N. Mahmudov
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  • 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.

    Mathematics Subject Classification: Primary: 34A60, 34G25; Secondary: 65K05, 90C25.


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  • [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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