# American Institute of Mathematical Sciences

April  2007, 3(2): 399-413. doi: 10.3934/jimo.2007.3.399

## Linear programming solutions of periodic optimization problems: approximation of the optimal control

 1 Centre for Industrial and Applicable Mathematics, University of South Australia, Mawson Lakes, SA 5095, Australia, Australia 2 WorkCover Corporation of South Australia, 100 Waymouth St., Adelaide SA 5000, Australia

Received  October 2006 Revised  January 2007 Published  April 2007

Deterministic long run average problems of optimal control are ''asymptotically equivalent" to infinite-dimensional linear programming problems ( LPP ) and the latter are approximated by finite dimensional LPP. The solutions of this finite dimensional LPP can be used for numerical analysis of periodic optimization problems. In the present paper we establish the convergence of controls constructed on the basis of the solution of the finite dimensional LPP to the optimal control of a periodic optimization problem. Results are illustrated with a numerical example.
Citation: Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399
 [1] Enkhbat Rentsen, J. Zhou, K. L. Teo. A global optimization approach to fractional optimal control. Journal of Industrial & Management Optimization, 2016, 12 (1) : 73-82. doi: 10.3934/jimo.2016.12.73 [2] Satoshi Ito, Soon-Yi Wu, Ting-Jang Shiu, Kok Lay Teo. A numerical approach to infinite-dimensional linear programming in $L_1$ spaces. Journal of Industrial & Management Optimization, 2010, 6 (1) : 15-28. doi: 10.3934/jimo.2010.6.15 [3] Vladimir Gaitsgory, Ilya Shvartsman. Linear programming estimates for Cesàro and Abel limits of optimal values in optimal control problems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021102 [4] Gengsheng Wang, Guojie Zheng. The optimal control to restore the periodic property of a linear evolution system with small perturbation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1621-1639. doi: 10.3934/dcdsb.2010.14.1621 [5] Vladimir Gaitsgory, Alex Parkinson, Ilya Shvartsman. Linear programming based optimality conditions and approximate solution of a deterministic infinite horizon discounted optimal control problem in discrete time. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1743-1767. doi: 10.3934/dcdsb.2018235 [6] Vladimir Gaitsgory, Alex Parkinson, Ilya Shvartsman. Linear programming formulations of deterministic infinite horizon optimal control problems in discrete time. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3821-3838. doi: 10.3934/dcdsb.2017192 [7] Sebastian Engel, Karl Kunisch. Optimal control of the linear wave equation by time-depending BV-controls: A semi-smooth Newton approach. Mathematical Control & Related Fields, 2020, 10 (3) : 591-622. doi: 10.3934/mcrf.2020012 [8] Tijana Levajković, Hermann Mena, Amjad Tuffaha. The stochastic linear quadratic optimal control problem in Hilbert spaces: A polynomial chaos approach. Evolution Equations & Control Theory, 2016, 5 (1) : 105-134. doi: 10.3934/eect.2016.5.105 [9] Rein Luus. Optimal control of oscillatory systems by iterative dynamic programming. Journal of Industrial & Management Optimization, 2008, 4 (1) : 1-15. doi: 10.3934/jimo.2008.4.1 [10] Evelyn Herberg, Michael Hinze, Henrik Schumacher. Maximal discrete sparsity in parabolic optimal control with measures. Mathematical Control & Related Fields, 2020, 10 (4) : 735-759. doi: 10.3934/mcrf.2020018 [11] Jorge San Martín, Takéo Takahashi, Marius Tucsnak. An optimal control approach to ciliary locomotion. Mathematical Control & Related Fields, 2016, 6 (2) : 293-334. doi: 10.3934/mcrf.2016005 [12] K. Renee Fister, Jennifer Hughes Donnelly. Immunotherapy: An Optimal Control Theory Approach. Mathematical Biosciences & Engineering, 2005, 2 (3) : 499-510. doi: 10.3934/mbe.2005.2.499 [13] Yibing Lv, Zhongping Wan. Linear bilevel multiobjective optimization problem: Penalty approach. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1213-1223. doi: 10.3934/jimo.2018092 [14] Torsten Trimborn, Lorenzo Pareschi, Martin Frank. Portfolio optimization and model predictive control: A kinetic approach. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6209-6238. doi: 10.3934/dcdsb.2019136 [15] Yi Zhang, Yong Jiang, Liwei Zhang, Jiangzhong Zhang. A perturbation approach for an inverse linear second-order cone programming. Journal of Industrial & Management Optimization, 2013, 9 (1) : 171-189. doi: 10.3934/jimo.2013.9.171 [16] Shiyun Wang, Yong-Jin Liu, Yong Jiang. A majorized penalty approach to inverse linear second order cone programming problems. Journal of Industrial & Management Optimization, 2014, 10 (3) : 965-976. doi: 10.3934/jimo.2014.10.965 [17] Radoslaw Pytlak. Numerical procedure for optimal control of higher index DAEs. Discrete & Continuous Dynamical Systems, 2011, 29 (2) : 647-670. doi: 10.3934/dcds.2011.29.647 [18] Emmanuel Trélat. Optimal control of a space shuttle, and numerical simulations. Conference Publications, 2003, 2003 (Special) : 842-851. doi: 10.3934/proc.2003.2003.842 [19] G. Machado, L. Trabucho. Analytical and numerical solutions for a class of optimization problems in elasticity. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1013-1032. doi: 10.3934/dcdsb.2004.4.1013 [20] Radu Ioan Boţ, Anca Grad, Gert Wanka. Sequential characterization of solutions in convex composite programming and applications to vector optimization. Journal of Industrial & Management Optimization, 2008, 4 (4) : 767-782. doi: 10.3934/jimo.2008.4.767

2019 Impact Factor: 1.366