Optimization of the Lagrange problem with second order discrete and differential inclusions and spatial boundary conditions

Abstract

In this paper, we derive optimality conditions for the Lagrange problem with second-order differential inclusions (DFIs) and spatial boundary conditions. The Lagrangian and the set-valued mapping here also depend on the derivative of the sought trajectory. The difficulties that arise here are related to the construction of the adjoint discrete and differential inclusions. The novelty here lies in using the discretized method to establish the optimality condition for both discrete and DFIs. Optimality conditions for the discrete problem are generated by applying the concept locally adjoint mapping(LAM). Equivalence theorems are used to obtain the so-called Mahmudov's adjoint conditions for the discrete-approximat problem. Moreover, passing to the limit, we get sufficient optimality conditions for the continuous problem. Unlike the Euler-Lagrange DFI, which only provides first-order optimality conditions, Mahmudov's adjoint inclusion is a powerful tool for establishing optimality conditions for higher-order problems.  The findings are reinforced with an example. We also obtain similar results for the non-convex problem by using the concept of local tents.

Keywords

• discrete inclusion
• differential inclusions
• spatial boundary conditions
• discrete-approximation
• optimization

References

1. [1] N.U. Ahmed and K.L. Teo, Optimal Control of Distributed Parameter Systems, Elsevier Sci. Inc., New York, 1981.
2. [2] G. Cicek and E.N. Mahmudov, The Problem of Mayer for discrete and differential inclusions with initial boundary constraints, Appl. Math. Inf. Sci. 10 (5), 1719-1728, 2016.
3. [3] G. Cicek and E.N. Mahmudov, Optimization of Mayer functional in problems with discrete and differential inclusions and viability constraints, Turk. J. Math. 45 (5), 2084-2102, 2021.
4. [4] S. Demir Sağlam and E.N. Mahmudov, On duality in convex optimization of secondorder differential inclusions with periodic boundary conditions, Hacet. J. Math. Stat. 51(6), 1588-1599, 2022.
5. [5] S. Demir Sağlam and E.N. Mahmudov, Convex optimization of nonlinear inequality with higher order derivatives, Appl. Anal. 102 (5), 1473-1489, 2023.
6. [6] I.M. Gelfand and S.V. Fomin, Calculus of Variations, Dover Publ., 2000.
7. [7] W. M. Haddad, V.S. Chellaboina, J.L. Fausz and C. Abdallah, Optimal discrete-time control for non-linear cascade systems, J. of the Frankl. Instit. 335 (5), 827-839, 1998.
8. [8] M. Kisielewicz, Differential Inclusions and Optimal Control, Kluwer Acad. Publ., London, 1991.

9. [9] P. D. Loewen and R. T. Rockafellar, Optimal control of unbounded differential inclusions, SIAM J. Control Optim. 32, 442-470, 1994.
10. [10] E.N. Mahmudov, Approximation and Optimization of Discrete and Differential Inclusions, Elsevier, Boston, USA, 2011.
11. [11] E.N. Mahmudov,Approximation and optimization of higher order discrete and differential inclusions, Nonlinear Differ. Equ. Appl.(NoDEA) 21 (1), 1–26, 2014.
12. [12] E.N. Mahmudov, Optimization of second order Discrete approximation inclusions, Numer. Funct. Anal. and Optim. 36(5), 624-643, 2015.
13. [13] E.N. Mahmudov, Convex optimization of second order discrete and differential inclusions with inequality constraints, J. Convex Anal. 25, 1-26, 2018.
14. [14] E.N. Mahmudov, Optimal control of evolution differential inclusions with polynomial linear differential operators, Evol. Equ. Control Theory 8 (3), 603-619, 2019.
15. [15] E.N. Mahmudov,Optimal control of higher order differential inclusions with functional constraints, ESAIM Control Optim. Calc. Var. 26 (37), 23, 2020.
16. [16] E.N. Mahmudov, Optimization of fourth order Sturm-Liouville type differential inclusions with initial point constraints, J. Ind. Manag. Optim. 16(1), 169-187, 2020.
17. [17] E.N. Mahmudov, Optimization of Higher-Order Differential Inclusions with Special Boundary Value Conditions, J. Optim. Theory Appl. 192, 36-55, 2022.
18. [18] E.N. Mahmudov and G. Cicek, Optimization of differential inclusions of Bolza type and dualities, Appl. Comput. Math. 6 (1), 88-96, 2007.
19. [19] E. N. Mahmudov and D. I. Mastaliyeva, Optimal Control Of Higher-Order Retarded Differential Inclusions, J. Indust. Manag. Optim. 19 (9), 6544-6557, 2023.
20. [20] B.S. Mordukhovich, Variational Analysis and Generalized Differentiation, I: Basic Theory; II: Applications, Springer, Grundlehren Series, Fundamental Principles of Mathematical Sciences , Vol. 330 and 331, 2006.
21. [21] B.S. Mordukhovich and T. Pennanen, Epi-convergent discretization of the generalized Bolza problem in dynamic optimization, Optim. Lett. 1, 379-390, 2007.
22. [22] L.S. Pontryagin, V.G. Boltyanskii , R.V. Gamkrelidze and E.F. Mishchenko, The Mathematical theory of optimal processes, John Wiley Sons Inc., New York, 1965.
23. [23] R.T. Rockafellar, Convex Analysis, Princet. Univ. Press, Princeton, NewJersey, 1997.
24. [24] Z. Zhou, X. Yang and Q. Qiu, Optimality conditions of the set-valued optimization problem with generalized cone convex set-valued maps characterized by contingent epiderivative, Acta Math. Appl. Sin. Engl. Ser. 34, 11-18, 2018.