Optimal Control Applications in the Study of Production Management



optimal control, automatic control, controllability, production management, Pontryagin Maximum Principle, Lie algebra


A mathematical model for an economic problem of production management is proposed. The continuous optimal control problem is solved, by using the Pontryagin Maximum Principle at the level of a new space, called Lie algebroid. The controllability of the economic system is studied by using Lie geometric methods and involves restrictions on the final stock quantities. Finally, a numerical application is given.


Agrachev, A.; Sachkov, Y. (2004). Control Theory from the Geometric Viewpoint. Encyclopedia of Mathematical Sciences 87, Springer, 2004. https://doi.org/10.1007/978-3-662-06404-7

Brocket, R. (1973). Lie algebra and Lie groups in control theory. Geometrical Methods in Control Theory, Springer, 43-82, 1973. https://doi.org/10.1007/978-94-010-2675-8_2

Caputo, M. (2005). Foundations of Dynamic Economic Analysis: Optimal Control Theory and Applications. Cambridge University Press, 2005. https://doi.org/10.1017/CBO9780511806827

Caruntu, C. F.; Velandia-Cardenas, C.; Liu, X.; Vargas, A. (2018). Model predictive control of stochastic linear systems with probability constraints. Int. J. Comp. Commun. Control, 13(6), 927-937, 2018. https://doi.org/10.15837/ijccc.2018.6.3383

Chazal, M.; Jouini, E.; Tahraoui, R. (2008). Production planning and inventories optimization: A backward approach in the convex storage cost case. J. Math. Economics, 44(9-10), 997-1023, 2008. https://doi.org/10.1016/j.jmateco.2007.04.011

Chitsaz, H.; LaValle, S. M.; Balkcom, D.; Mason, M. (2009). Minimum Wheel-Rotation Paths for Differential-Drive Mobile Robots. The International Journal of Robotics Research, 28(1), 66-80, 2009. https://doi.org/10.1177/0278364908096750

Dubins, L.E. (1957). On curves of minimal length with a constraint on average curvature and with prescribed initial and terminal positions and tangents. American Journal of Mathematics, 79, 497-516, 1957. https://doi.org/10.2307/2372560

Feichtinger, G.; Hartl, R. (1985). Optimal pricing and production in an inventory model. European J. Operational Research, 19(1), 45-56, 1985. https://doi.org/10.1016/0377-2217(85)90307-8

Gayon, J. P.; Vercraene, S.; Flapper, S. D. (2017). Optimal control of a production-inventory system with product returns and two disposal options. European J. Operational Research, 262(2), 499-508, 2017. https://doi.org/10.1016/j.ejor.2017.03.018

Isidori, A. (1995). Nonlinear Control Systems. Springer, 1995. https://doi.org/10.1007/978-1-84628-615-5

Jurdjevic, V. (2008). Geometric Control Theory. Cambridge Studies in Advanced Mathematics 52, Cambridge University Press, 2008.

Kamien, M.I.; Schwartz, N.L. (2006). Dynamic optimization. Elsevier, 2006.

LaValle, S. M. (2006). Planning Algorithms. Cambridge University Press, 2006. https://doi.org/10.1017/CBO9780511546877

Liu, R.; Liu, G. (2018). Maximum principle for a nonlinear size-structured model of fish and fry management. Nonlinear Analysis: Modelling and Control, 23(4), 533-552, 2018. https://doi.org/10.15388/NA.2018.4.5

Maccini, L. J.; Moore, B.; Schaller, H. (2015). Inventory behavior with permanent sales stocks. J. Economic Dynamics and Control, 53, 290-313, 2015. https://doi.org/10.1016/j.jedc.2015.02.010

Martinez, E. (2004). Reduction in optimal control theory. Rep. Math. Phys., 53(1), 79-90, 2004. https://doi.org/10.1016/S0034-4877(04)90005-5

Mackenzie, K. (1987). Lie Groupoids and Lie Algebroids in Differential Geometry. London Mathematical Society Lecture Note Series 124, Cambridge University Press, 1987. https://doi.org/10.1017/CBO9780511661839

Olsson, F. (2019). Simple modeling techniques for base-stock inventory systems with state dependent demand rates. Mathematical Methods of Operations Research, 90, 61-76, 2019. https://doi.org/10.1007/s00186-018-0654-0

Ortega, M.; Lin, L. (2004). Control theory applications to the production-inventory problem: a review. International Journal of Production Research, 42(11), 2303-2322, 2004. https://doi.org/10.1080/00207540410001666260

Popescu, L. (2009). Lie algebroids framework for distributional systems. Annals Univ. Al. I. Cuza, Iasi, series I, Mathematics, 55(2), 257-274, 2009.

Popescu, L. (2019). Applications of driftless control affine systems to a problem of inventory and production. Studies in Informatics and Control, 28(1), 25-34, 2019. https://doi.org/10.24846/v28i1y201903

Sethi, S. P.; Thompson, G. L. (2000). Optimal Control Theory: Applications to Management Science and Economics. Springer, 2000.

Weber, T.A. (2011). Optimal Control Theory with Applications in Economics. MIT Press, 2011. https://doi.org/10.7551/mitpress/9780262015738.001.0001



Most read articles by the same author(s)

Obs.: This plugin requires at least one statistics/report plugin to be enabled. If your statistics plugins provide more than one metric then please also select a main metric on the admin's site settings page and/or on the journal manager's settings pages.