# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2018145

## Optimization of fourth order Sturm-Liouville type differential inclusions with initial point constraints

 1 Department of Mathematics, Istanbul Technical University, Turkey 2 Azerbaijan National Academy of Sciences, Institute of Control Systems, Azerbaijan

* Corresponding author: Elimhan N. Mahmudov

Received  July 2017 Revised  May 2018 Published  September 2018

The present paper studies a new class of problems of optimal control theory with differential inclusions described by fourth order Sturm-Liouville type differential operators (SLDOs). Then, there arises a rather complicated problem with simultaneous determination of the SLDOs with variable coefficients and a Mayer functional depending of high order derivatives of searched functions. The sufficient conditions, containing both the Euler-Lagrange and Hamiltonian type inclusions and "transversality" conditions are derived. Formulation of the transversality conditions at the endpoints $t = 0$ and $t = 1$ of the considered time interval plays a substantial role in the next investigations without which it is hardly ever possible to get any optimality conditions. The main idea of the proof of optimality conditions of Mayer problem for differential inclusions with fourth order SLDO is the use of locally-adjoint mappings. The method is demonstrated in detail as an example for the semilinear optimal control problem, for which the Weierstrass-Pontryagin maximum principle is obtained.

Citation: Elimhan N. Mahmudov. Optimization of fourth order Sturm-Liouville type differential inclusions with initial point constraints. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018145
##### References:
 [1] N. U. Ahmed, Differential inclusions operator valued measures and optimal control, Dynamic Systems and Applications, 16 (2007), 13-35. [2] A. Auslender and J. Mechler, Second order viability problems for differential inclusions, Journal of Mathematical Analysis and Applications, 181 (1994), 205-218. doi: 10.1006/jmaa.1994.1015. [3] D. Azzam-Laouir, S. Lounis and L. Thibault, Existence solutions for second-order differential inclusions with nonconvex perturbations, Applicable Analysis, 86 (2011), 1199-1210. doi: 10.1080/00036810701460511. [4] R. Burachik, J. Lopes and G. Silva, An inexact interior point proximal method for the variational inequality problem, Computational and Applied Mathematics, 28 (2009), 15-36. doi: 10.1590/S0101-82052009000100002. [5] F. H. Clarke, Functional Analysis, Calculus of Variations and Optimal Control, Graduate Texts in Mathematics, Springer, 2013. doi: 10.1007/978-1-4471-4820-3. [6] M. Fukushima, A class of gap functions for quasi-variational inequality problems, Journal of Industrial and Management Optimization, 3 (2007), 165-171. doi: 10.3934/jimo.2007.3.165. [7] Q. Liqun, K. L. Teo and X. Yang, Optimization and Control with Applications, Springer, 2005. doi: 10.1007/b104943. [8] S. J. Li, S. K. Zhu and K. L. Teo, New generalized second-order contingent epiderivatives and set-valued optimization problems, Journal of Optimization Theory and Applications, 152 (2012), 587-604. doi: 10.1007/s10957-011-9915-2. [9] Y. Liu, J. Wu and Z. Li, Impulsive boundary value problems for Sturm-Liouville type differential inclusions, Journal of Systems Science and Complexity, 20 (2007), 370-380. doi: 10.1007/s11424-007-9032-3. [10] L. Liu, X. Zhang and Y. Wu, Positive solutions of Fourth-order nonlinear singular Sturm-Liouville eigenvalue problems, Journal of Mathematical Analysis and Applications, 326 (2007), 1212-1224. doi: 10.1016/j.jmaa.2006.03.029. [11] E. N. Mahmudov, Optimal control of cauchy problem for first-order discrete and partial differential inclusions, J. Dyn. Contr. Syst., 15 (2009), 587-610. doi: 10.1007/s10883-009-9073-0. [12] E. N. Mahmudov, Locally adjoint mappings and optimization of the first boundary value problem for hyperbolic type discrete and differential inclusions, Nonlin. Anal., 67 (2007), 2966-2981. doi: 10.1016/j.na.2006.09.054. [13] E. N. Mahmudov, Necessary and sufficient conditions for discrete and differential inclusions of elliptic type, J. Math. Anal. Appl., 323 (2006), 768-789. doi: 10.1016/j.jmaa.2005.10.069. [14] E. N. Mahmudov, Approximation and Optimization of Discrete and Differential Inclusions Elsevier, 2011. doi: 10.1016/B978-0-12-388428-2.00001-1. [15] 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. [16] E. N. Mahmudov, Transversality condition and optimization of higher order ordinary differential inclusions, Optimization, 64 (2005), 2131-2144. doi: 10.1080/02331934.2014.929681. [17] E. N. Mahmudov, Mathematical programming and polyhedral optimization of second order discrete and differential inclusions, Pacific J. Optim., 11 (2015), 511-525. [18] E. N. Mahmudov, Optimization of second order discrete approximation inclusions, Numerical Functional Analysis and Optimizations, 36 (2015), 624-643. doi: 10.1080/01630563.2015.1014048. [19] E. N. Mahmudov, Single Variable Differential and Integral Calculus, Atlantis Press-Springer, 2013. doi: 10.2991/978-94-91216-86-2. [20] B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, I: Basic Theory; II: Applications, Grundlehren Series (Fundamental Principles of Mathematical Sciences), Vol. 330 and 331, Springer, 2006. [21] D. G. Long, D. Q. A and V. T. Luan, Iterative method for solving a fourth-order differential equation with nonlinear boundary condition, Applied Mathematical Sciences, 4 (2010), 3467-3481. [22] S. H. Saker, R. P. Agarwal and D. O'Regan, Properties of solutions of fourth-order differential equations with boundary conditions, Journal of Inequalities and Applications, 2013 (2013), 15pp. doi: 10.1186/1029-242X-2013-278. [23] M. Thera, S. Adly and A. Hantoute, Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain, Mathematical Programming, 157 (2016), 349-374. doi: 10.1007/s10107-015-0938-6. [24] Y. Xu and Z. Peng, Higher-order sensitivity analysis in set-valued optimization under Henig efficiency, Journal of Industrial and Management Optimization, 13 (2017), 313-327. doi: 10.3934/jimo.2016019. [25] 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, Journal of Industrial and Management Optimization, 11 (2014), 673-683. doi: 10.3934/jimo.2015.11.673.

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##### References:
 [1] N. U. Ahmed, Differential inclusions operator valued measures and optimal control, Dynamic Systems and Applications, 16 (2007), 13-35. [2] A. Auslender and J. Mechler, Second order viability problems for differential inclusions, Journal of Mathematical Analysis and Applications, 181 (1994), 205-218. doi: 10.1006/jmaa.1994.1015. [3] D. Azzam-Laouir, S. Lounis and L. Thibault, Existence solutions for second-order differential inclusions with nonconvex perturbations, Applicable Analysis, 86 (2011), 1199-1210. doi: 10.1080/00036810701460511. [4] R. Burachik, J. Lopes and G. Silva, An inexact interior point proximal method for the variational inequality problem, Computational and Applied Mathematics, 28 (2009), 15-36. doi: 10.1590/S0101-82052009000100002. [5] F. H. Clarke, Functional Analysis, Calculus of Variations and Optimal Control, Graduate Texts in Mathematics, Springer, 2013. doi: 10.1007/978-1-4471-4820-3. [6] M. Fukushima, A class of gap functions for quasi-variational inequality problems, Journal of Industrial and Management Optimization, 3 (2007), 165-171. doi: 10.3934/jimo.2007.3.165. [7] Q. Liqun, K. L. Teo and X. Yang, Optimization and Control with Applications, Springer, 2005. doi: 10.1007/b104943. [8] S. J. Li, S. K. Zhu and K. L. Teo, New generalized second-order contingent epiderivatives and set-valued optimization problems, Journal of Optimization Theory and Applications, 152 (2012), 587-604. doi: 10.1007/s10957-011-9915-2. [9] Y. Liu, J. Wu and Z. Li, Impulsive boundary value problems for Sturm-Liouville type differential inclusions, Journal of Systems Science and Complexity, 20 (2007), 370-380. doi: 10.1007/s11424-007-9032-3. [10] L. Liu, X. Zhang and Y. Wu, Positive solutions of Fourth-order nonlinear singular Sturm-Liouville eigenvalue problems, Journal of Mathematical Analysis and Applications, 326 (2007), 1212-1224. doi: 10.1016/j.jmaa.2006.03.029. [11] E. N. Mahmudov, Optimal control of cauchy problem for first-order discrete and partial differential inclusions, J. Dyn. Contr. Syst., 15 (2009), 587-610. doi: 10.1007/s10883-009-9073-0. [12] E. N. Mahmudov, Locally adjoint mappings and optimization of the first boundary value problem for hyperbolic type discrete and differential inclusions, Nonlin. Anal., 67 (2007), 2966-2981. doi: 10.1016/j.na.2006.09.054. [13] E. N. Mahmudov, Necessary and sufficient conditions for discrete and differential inclusions of elliptic type, J. Math. Anal. Appl., 323 (2006), 768-789. doi: 10.1016/j.jmaa.2005.10.069. [14] E. N. Mahmudov, Approximation and Optimization of Discrete and Differential Inclusions Elsevier, 2011. doi: 10.1016/B978-0-12-388428-2.00001-1. [15] 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. [16] E. N. Mahmudov, Transversality condition and optimization of higher order ordinary differential inclusions, Optimization, 64 (2005), 2131-2144. doi: 10.1080/02331934.2014.929681. [17] E. N. Mahmudov, Mathematical programming and polyhedral optimization of second order discrete and differential inclusions, Pacific J. Optim., 11 (2015), 511-525. [18] E. N. Mahmudov, Optimization of second order discrete approximation inclusions, Numerical Functional Analysis and Optimizations, 36 (2015), 624-643. doi: 10.1080/01630563.2015.1014048. [19] E. N. Mahmudov, Single Variable Differential and Integral Calculus, Atlantis Press-Springer, 2013. doi: 10.2991/978-94-91216-86-2. [20] B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, I: Basic Theory; II: Applications, Grundlehren Series (Fundamental Principles of Mathematical Sciences), Vol. 330 and 331, Springer, 2006. [21] D. G. Long, D. Q. A and V. T. Luan, Iterative method for solving a fourth-order differential equation with nonlinear boundary condition, Applied Mathematical Sciences, 4 (2010), 3467-3481. [22] S. H. Saker, R. P. Agarwal and D. O'Regan, Properties of solutions of fourth-order differential equations with boundary conditions, Journal of Inequalities and Applications, 2013 (2013), 15pp. doi: 10.1186/1029-242X-2013-278. [23] M. Thera, S. Adly and A. Hantoute, Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain, Mathematical Programming, 157 (2016), 349-374. doi: 10.1007/s10107-015-0938-6. [24] Y. Xu and Z. Peng, Higher-order sensitivity analysis in set-valued optimization under Henig efficiency, Journal of Industrial and Management Optimization, 13 (2017), 313-327. doi: 10.3934/jimo.2016019. [25] 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, Journal of Industrial and Management Optimization, 11 (2014), 673-683. doi: 10.3934/jimo.2015.11.673.
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