On duality in convex optimization of second-order differential inclusions with periodic boundary conditions

Year 2022, , 1588 - 1599, 01.12.2022

Sevilay Demir Sağlam , Elimhan Mahmudov

https://doi.org/10.15672/hujms.1056259

Cited By: 2

Abstract

The present paper is devoted to the duality theory for the convex optimal control problem of second-order differential inclusions with periodic boundary conditions. First, we use an auxiliary problem with second-order discrete-approximate inclusions and focus on formulating sufficient conditions of optimality for the differential problem. Then, we concentrate on the duality that exists in periodic boundary conditions to establish a dual problem for the differential problem and prove that Euler-Lagrange inclusions are duality relations for both primal and dual problems. Finally, we consider an example of the duality for the second-order linear optimal control problem.

Keywords

differential inclusion, optimality conditions, duality, transversality condition

References

• [1] R.P. Agarwal and B. Ahmad, Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions, Comput. Math. with Appl. 62 (3), 1200-1214, 2011.
• [2] R.I. Bot, Conjugate Duality in Convex Optimization, Springer-Verlag, Berlin, 2010.
• [3] R.S. Burachik and V. Jeyakumar, A dual condition for the convex subdifferential sum formula with applications, J. Convex Anal. 12 (2), 279-290, 2005.
• [4] F.H. Clarke, Functional Analysis, Calculus of Variations and Optimal Control, Graduate Texts in Mathematics, Springer, 2013.
• [5] S. Demir Sağlam, The optimality principle for second-order discrete and discrete approximate inclusions, Int. J. Optim. Control: Theor. Appl. 11 (2), 206-215, 2021.
• [6] S. Demir Sağlam and E.N. Mahmudov, Optimality conditions for higher-order polyhedral discrete and differential inclusions, Filomat, 34 (13), 4533-4553, 2020.
• [7] S. Demir Sağlam and E.N. Mahmudov, Polyhedral optimization of second-order discrete and differential inclusions with delay, Turkish J. Math. 45 (1), 244-263, 2021.
• [8] S. Demir Sağlam and E.N. Mahmudov, Convex optimization of nonlinear inequality with higher order derivatives, Appl. Anal. doi:10.1080/00036811.2021.1988578, 2021.
• [9] S. Demir Sağlam and E.N. Mahmudov, The Lagrange Problem for differential inclusions with boundary value conditions and duality, Pac. J. Optim. 17 (2), 209-225, 2021.
• [10] S. Demir Sağlam and E.N. Mahmudov, Duality problems with second-order polyhedral discrete and differential inclusions, Bull. Iran. Math.Soc. 48 (2), 537-562, 2022.
• [11] I. Ekeland and R. Temam, Analyse convex et problemes variationelles, Dunod and Gauthier Villars, Paris, 1974.
• [12] A. Hamidoglu, Null controllability of heat equation with switching controls under Robin’s boundary condition, Hacet. J. Math. Stat. 45 (2), 373-379, 2016.
• [13] X. Li, M. Bohner and C.-K.Wang, Impulsive differential equations: Periodic solutions and applications, Automatica, 52, 173-178, 2015.
• [14] X. Liu, Y.B. Zhang and H.P. Shi, Existence and nonexistence results for a fourth-order discrete Dirichlet boundary value problem, Hacet. J. Math. Stat. 44 (4), 855-866, 2015.
• [15] E.N. Mahmudov, Approximation and Optimization of Discrete and Differential Inclusions, Elsevier, Boston, USA, 2011.
• [16] E.N. Mahmudov, Optimization of Second Order Discrete Approximation Inclusions, Numer. Funct. Anal. Optim. 36 (5), 624-643, 2015.
• [17] E.N. Mahmudov, Optimization of Higher-Order Differential Inclusions with Endpoint Constraints and Duality, Adv. Mathem. Models Appl. 6 (1), 5-21, 2021.
• [18] E.N. Mahmudov, Infimal Convolution and Duality in Problems with Third-Order Discrete and Differential Inclusions, J. Optim. Theory Appl. 184, 781-809, 2020.
• [19] E.N. Mahmudov, Optimal control of high order viable differential inclusions and duality, Appl. Anal. 101 (7) , 2616-2635, 2022
• [20] M.J. Mardanov, T.K. Melikov, S.T. Malik and K. Malikov, First- and second-order necessary conditions with respect to components for discrete optimal control problems, J. Comput. Appl. Math. 364, 112342, 2020.
• [21] 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, Berlin, 2006.
• [22] Z. Pavi´c and V. Novoselac, Investigating an overdetermined system of linear equations by using convex functions, Hacet. J. Math. Stat. 46 (5), 865-874, 2017.
• [23] K.N. Soltanov, On Semi-Continuous Mappings, Equations and Inclusions in a Banach Space, Hacet. J. Math. Stat. 37 (1), 9-24, 2000.