# American Institute of Mathematical Sciences

• Previous Article
Convergence of interval AOR method for linear interval equations
• NACO Home
• This Issue
• Next Article
A new inertial-projection algorithm for approximating common solution of variational inequality and fixed point problems of multivalued mappings
June  2022, 12(2): 279-291. doi: 10.3934/naco.2021005

## Optimal control of a dynamical system with intermediate phase constraints and applications in cash management

 1 Department of Mathematics and Computer Science, University of Mila, 043000 Mila, Algeria, Research Unit LaMOS, University of Bejaia 2 Research Unit LaMOS, Department of Operational Research, University of Bejaia, 06000 Bejaia, Algeria

Received  May 2020 Revised  January 2021 Published  June 2022 Early access  February 2021

The aim of this work is to apply the results of R. Gabasov et al. [4,14] to an extended class of optimal control problems in the Bolza form, with intermediate phase constraints and multivariate control. In this paper, the developed iterative numerical method avoids the discretization of the dynamical system. Indeed, by using a piecewise constant control, the problem is reduced for each iteration to a linear programming problem, this auxiliary task allows to improve the value of the quality criterion. The process is repeated until the optimal or the suboptimal control is obtained. As an application, we use this method to solve an extension of the deterministic optimal cash management model of S.P. Sethi [31,32]. In this extension, we assume that the bank overdrafts and short selling of stock are allowed, but within the authorized time limit. The results of the numerical example show that the optimal decision for the firm depends closely on the intermediate moment, the optimal decision for the firm is to purchase until a certain date the stocks at their authorized maximum value in order to take advantage of the returns derived from stock. After that, it sales the stocks at their authorized maximum value in order to satisfy the constraint at the intermediate moment.

Citation: Mourad Azi, Mohand Ouamer Bibi. Optimal control of a dynamical system with intermediate phase constraints and applications in cash management. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 279-291. doi: 10.3934/naco.2021005
##### References:
 [1] A. V. Arutyunov and A. I. Okoulevitch, Necessary optimality conditions for optimal control problems with itermediate constraints, Journal of Dynamical and Control Systems, 4 (1998), 49-58.  doi: 10.2307/2152750. [2] M. Azi and M. O. Bibi, Optimal cash management with intermediate phase constraints, In Proc. of the International Conference on Financial mathematics Tools and Applications (MFOA'2019) University of Bejaia, Octobre 28-29, (2019), 14 – 23. doi: 10.2307/2152750. [3] M. Azi and M. O. Bibi, Optimal Control of Linear Dynamical System with Intermediate Phase Constraints, In Proc. of the 11th Conference on the Optimization and Information Systems, (COSI'2014), (2014), 347–356. doi: 10.2307/2152750. [4] N. V. Balashevich, R. Gabasov and F. M. Kirillova, Algorithms for open loop and closed loop optimization of control systems with intermediate phase constraints, Zh. Vychisl. Mat. Fiz, 41 (2001), 1485-1504.  doi: 10.2307/2152750. [5] M. O. Bibi, Methods for Solving Linear-Quadratic Problems of Optimal Control, Ph.D Thesis, University of Minsk, 1985. [6] M. O. Bibi, Optimization of a linear dynamic system with double terminal constraint on the trajectories, Optimization, 30 (1994), 359-366.  doi: 10.2307/2152750. [7] M. O. Bibi, Support method for solving a linear-quadratic problem with polyhedral constraints on control, Optimization, 37 (1996), 139-147.  doi: 10.2307/2152750. [8] M. O. Bibi and M. Bentobache, A hybrid direction algorithm for solving linear programs, International Journal of Computer Mathematics, 92 (2015), 201-216.  doi: 10.2307/2152750. [9] M. O. Bibi and S. Medjdoub, Optimal control of a linear-quadratic problem with free initial condition, In Proc. 26th European conference on operational research, Rome, Italy, (2013), 362–362. doi: 10.2307/2152750. [10] T. Bjork, M. H. A. Davis and C. Landen, Optimal investment under partial information, Mathematical Methods of Operations Research, 71 (2010), 371-399.  doi: 10.2307/2152750. [11] W. J. Baumol, The transactions demand for cash: An inventory theoretic approach, Quarterly Journal of Economics, 66 (1952), 545-556.  doi: 10.2307/2152750. [12] M. N. Dmitruk and R. Gabasov, The optimal policy of dividends, investments, and capital distribution for the dynamic model of a company, Automation and Remote Control, 62 (2001), 1349-1365.  doi: 10.2307/2152750. [13] A. V. Dmitruk and A. M. Kaganovich, Maximum principle for optimal control problems with intermediate constraints, Computational Mathematics and Modeling, 22 (2011), 180-215.  doi: 10.2307/2152750. [14] L. D. Erovenko, Algorithm for optimization of a non-stationary dynamic system, in Constructive Theory of Extremal Problems (eds. R. Gabasov and F.M. Kirillova), University Press, Minsk, (1984), 76–89. [15] R. Gabasov, N. V. Balashevich and F. M. Kirillova, Constructive methods of optimization of dynamical systems, Vietnam Journal of Mathematics, 30 (2002), 201-239.  doi: 10.2307/2152750. [16] R. Gabasov, M. N. Dmitruk and F. M. Kirillova, Optimization of the multidimensional control systems with parallelepiped constraints, Automation and Remote Control, 63 (2002), 345-366.  doi: 10.2307/2152750. [17] R. Gabasov, O. P. Grushevich and F. M. Kirillova, Optimal control of the delay linear systems with allowance for the terminal state constraints, Automation and Remote Control, 68 (2007), 2097-2112.  doi: 10.2307/2152750. [18] R. Gabasov, F. M. Kirillova and A. I. Tyatyushkin, Constructive Methods of Optimization, P.Ⅰ: Linear Problems, University Press, Minsk, 1984.  doi: 10.1007/978-1-4612-0873-0. [19] R. Gabasov and F. M. Kirillova, Constructive Methods of Optimization, P.Ⅱ: Control Problems, University Press, Minsk, 1984.  doi: 10.1007/978-1-4612-0873-0. [20] R. Gabasov, F. M. Kirillova, V. V. Alsevich, A. I. Kalinin, V. V. Krakhotko and N. S. Pavlenko, Methods of Optimization, Four Quarters, Minsk, 2011. doi: 10.1007/978-1-4612-0873-0. [21] R. Gabasov, F. M. Kirillova and N. S. Pavlenok, Constructing open-loop and closed-loop solutions of linear-quadratic optimal control problems, Computational Mathematics and Mathematical Physics, 48 (2008), 1715-1745. doi: 10.2307/2152750. [22] R. Gabasov, F. M. Kirillova and S. V. Prischepova, Optimal Feedback Control, Springer-Verlag, London, 1995. doi: 10.1007/978-1-4612-0873-0. [23] F. Ghellab and M. O. Bibi, Optimality and suboptimality criteria in a quadratic problem of optimal control with a piecewise linear entry, International Journal of Mathematics in Operational Research, 2020. doi: 10.2307/2152750. [24] O. Hilton, P. M. Kort and P. J. J. M. Loon, Dynamic Policies of a Firm: An Optimal Control Approach, Springer, Berlin, 1993. doi: 10.1007/978-1-4612-0873-0. [25] N. Khimoum and M. O. Bibi, Primal-dual method for solving a linear-quadratic multi-input optimal control problem, Optimization Letters, 14 (2020), 653-669.  doi: 10.2307/2152750. [26] R. Korn, Some applications of impulse control in mathematical finance, Mathematical Methods of Operations Research, 50 (1999), 493-518.  doi: 10.2307/2152750. [27] K. Li, E. Feng and Z. Xiu, Optimal control and optimization algorithm of nonlinear impulsive delay system producing 1, 3-Propanediol, Journal of Applied Mathematics and Computing, 24 (2007), 387-397.  doi: 10.2307/2152750. [28] W. I. Nathanson, Control Problems with intermediate constraints: A sufficient condition, Journal of Optimization Theory and Applications, 29 (1979), 253-290.  doi: 10.2307/2152750. [29] W. I. Nathanson, Control problems with intermediate constraints, Journal of Optimization Theory and Applications, 8 (1971), 256-270.  doi: 10.2307/2152750. [30] L. S. Pontryaguine, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, John Wiley and Sons, New Jersey, 1962. doi: 10.1007/978-1-4612-0873-0. [31] S. P. Sethi, A. Bensoussan and A. Chutani, Optimal cash management under uncertainty, Operations Research Letters, 37 (2009), 425-429.  doi: 10.2307/2152750. [32] S. P. Sethi, Optimal Control Theory: Applications to Management Sciences and Economics, Third edition, Springer Nature Switzerland, 2019. doi: 10.1007/978-1-4612-0873-0. [33] S. P. Sethi and Q. Zhang, Systems and Control: Foundations and Applications, Birkhauser Boston, 1994. doi: 10.1007/978-1-4612-0873-0.

show all references

##### References:
 [1] A. V. Arutyunov and A. I. Okoulevitch, Necessary optimality conditions for optimal control problems with itermediate constraints, Journal of Dynamical and Control Systems, 4 (1998), 49-58.  doi: 10.2307/2152750. [2] M. Azi and M. O. Bibi, Optimal cash management with intermediate phase constraints, In Proc. of the International Conference on Financial mathematics Tools and Applications (MFOA'2019) University of Bejaia, Octobre 28-29, (2019), 14 – 23. doi: 10.2307/2152750. [3] M. Azi and M. O. Bibi, Optimal Control of Linear Dynamical System with Intermediate Phase Constraints, In Proc. of the 11th Conference on the Optimization and Information Systems, (COSI'2014), (2014), 347–356. doi: 10.2307/2152750. [4] N. V. Balashevich, R. Gabasov and F. M. Kirillova, Algorithms for open loop and closed loop optimization of control systems with intermediate phase constraints, Zh. Vychisl. Mat. Fiz, 41 (2001), 1485-1504.  doi: 10.2307/2152750. [5] M. O. Bibi, Methods for Solving Linear-Quadratic Problems of Optimal Control, Ph.D Thesis, University of Minsk, 1985. [6] M. O. Bibi, Optimization of a linear dynamic system with double terminal constraint on the trajectories, Optimization, 30 (1994), 359-366.  doi: 10.2307/2152750. [7] M. O. Bibi, Support method for solving a linear-quadratic problem with polyhedral constraints on control, Optimization, 37 (1996), 139-147.  doi: 10.2307/2152750. [8] M. O. Bibi and M. Bentobache, A hybrid direction algorithm for solving linear programs, International Journal of Computer Mathematics, 92 (2015), 201-216.  doi: 10.2307/2152750. [9] M. O. Bibi and S. Medjdoub, Optimal control of a linear-quadratic problem with free initial condition, In Proc. 26th European conference on operational research, Rome, Italy, (2013), 362–362. doi: 10.2307/2152750. [10] T. Bjork, M. H. A. Davis and C. Landen, Optimal investment under partial information, Mathematical Methods of Operations Research, 71 (2010), 371-399.  doi: 10.2307/2152750. [11] W. J. Baumol, The transactions demand for cash: An inventory theoretic approach, Quarterly Journal of Economics, 66 (1952), 545-556.  doi: 10.2307/2152750. [12] M. N. Dmitruk and R. Gabasov, The optimal policy of dividends, investments, and capital distribution for the dynamic model of a company, Automation and Remote Control, 62 (2001), 1349-1365.  doi: 10.2307/2152750. [13] A. V. Dmitruk and A. M. Kaganovich, Maximum principle for optimal control problems with intermediate constraints, Computational Mathematics and Modeling, 22 (2011), 180-215.  doi: 10.2307/2152750. [14] L. D. Erovenko, Algorithm for optimization of a non-stationary dynamic system, in Constructive Theory of Extremal Problems (eds. R. Gabasov and F.M. Kirillova), University Press, Minsk, (1984), 76–89. [15] R. Gabasov, N. V. Balashevich and F. M. Kirillova, Constructive methods of optimization of dynamical systems, Vietnam Journal of Mathematics, 30 (2002), 201-239.  doi: 10.2307/2152750. [16] R. Gabasov, M. N. Dmitruk and F. M. Kirillova, Optimization of the multidimensional control systems with parallelepiped constraints, Automation and Remote Control, 63 (2002), 345-366.  doi: 10.2307/2152750. [17] R. Gabasov, O. P. Grushevich and F. M. Kirillova, Optimal control of the delay linear systems with allowance for the terminal state constraints, Automation and Remote Control, 68 (2007), 2097-2112.  doi: 10.2307/2152750. [18] R. Gabasov, F. M. Kirillova and A. I. Tyatyushkin, Constructive Methods of Optimization, P.Ⅰ: Linear Problems, University Press, Minsk, 1984.  doi: 10.1007/978-1-4612-0873-0. [19] R. Gabasov and F. M. Kirillova, Constructive Methods of Optimization, P.Ⅱ: Control Problems, University Press, Minsk, 1984.  doi: 10.1007/978-1-4612-0873-0. [20] R. Gabasov, F. M. Kirillova, V. V. Alsevich, A. I. Kalinin, V. V. Krakhotko and N. S. Pavlenko, Methods of Optimization, Four Quarters, Minsk, 2011. doi: 10.1007/978-1-4612-0873-0. [21] R. Gabasov, F. M. Kirillova and N. S. Pavlenok, Constructing open-loop and closed-loop solutions of linear-quadratic optimal control problems, Computational Mathematics and Mathematical Physics, 48 (2008), 1715-1745. doi: 10.2307/2152750. [22] R. Gabasov, F. M. Kirillova and S. V. Prischepova, Optimal Feedback Control, Springer-Verlag, London, 1995. doi: 10.1007/978-1-4612-0873-0. [23] F. Ghellab and M. O. Bibi, Optimality and suboptimality criteria in a quadratic problem of optimal control with a piecewise linear entry, International Journal of Mathematics in Operational Research, 2020. doi: 10.2307/2152750. [24] O. Hilton, P. M. Kort and P. J. J. M. Loon, Dynamic Policies of a Firm: An Optimal Control Approach, Springer, Berlin, 1993. doi: 10.1007/978-1-4612-0873-0. [25] N. Khimoum and M. O. Bibi, Primal-dual method for solving a linear-quadratic multi-input optimal control problem, Optimization Letters, 14 (2020), 653-669.  doi: 10.2307/2152750. [26] R. Korn, Some applications of impulse control in mathematical finance, Mathematical Methods of Operations Research, 50 (1999), 493-518.  doi: 10.2307/2152750. [27] K. Li, E. Feng and Z. Xiu, Optimal control and optimization algorithm of nonlinear impulsive delay system producing 1, 3-Propanediol, Journal of Applied Mathematics and Computing, 24 (2007), 387-397.  doi: 10.2307/2152750. [28] W. I. Nathanson, Control Problems with intermediate constraints: A sufficient condition, Journal of Optimization Theory and Applications, 29 (1979), 253-290.  doi: 10.2307/2152750. [29] W. I. Nathanson, Control problems with intermediate constraints, Journal of Optimization Theory and Applications, 8 (1971), 256-270.  doi: 10.2307/2152750. [30] L. S. Pontryaguine, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, John Wiley and Sons, New Jersey, 1962. doi: 10.1007/978-1-4612-0873-0. [31] S. P. Sethi, A. Bensoussan and A. Chutani, Optimal cash management under uncertainty, Operations Research Letters, 37 (2009), 425-429.  doi: 10.2307/2152750. [32] S. P. Sethi, Optimal Control Theory: Applications to Management Sciences and Economics, Third edition, Springer Nature Switzerland, 2019. doi: 10.1007/978-1-4612-0873-0. [33] S. P. Sethi and Q. Zhang, Systems and Control: Foundations and Applications, Birkhauser Boston, 1994. doi: 10.1007/978-1-4612-0873-0.
Optimal control $u_1^*(t)$
Optimal control $u_2^*(t)$
 [1] Piermarco Cannarsa, Hélène Frankowska, Elsa M. Marchini. On Bolza optimal control problems with constraints. Discrete and Continuous Dynamical Systems - B, 2009, 11 (3) : 629-653. doi: 10.3934/dcdsb.2009.11.629 [2] Leszek Gasiński. Optimal control problem of Bolza-type for evolution hemivariational inequality. Conference Publications, 2003, 2003 (Special) : 320-326. doi: 10.3934/proc.2003.2003.320 [3] Zhengyan Wang, Guanghua Xu, Peibiao Zhao, Zudi Lu. The optimal cash holding models for stochastic cash management of continuous time. Journal of Industrial and Management Optimization, 2018, 14 (1) : 1-17. doi: 10.3934/jimo.2017034 [4] Andrei V. Dmitruk, Alexander M. Kaganovich. Quadratic order conditions for an extended weak minimum in optimal control problems with intermediate and mixed constraints. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 523-545. doi: 10.3934/dcds.2011.29.523 [5] Hang-Chin Lai, Jin-Chirng Lee, Shuh-Jye Chern. A variational problem and optimal control. Journal of Industrial and Management Optimization, 2011, 7 (4) : 967-975. doi: 10.3934/jimo.2011.7.967 [6] Maria do Rosário de Pinho, Ilya Shvartsman. Lipschitz continuity of optimal control and Lagrange multipliers in a problem with mixed and pure state constraints. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 505-522. doi: 10.3934/dcds.2011.29.505 [7] Georg Vossen, Torsten Hermanns. On an optimal control problem in laser cutting with mixed finite-/infinite-dimensional constraints. Journal of Industrial and Management Optimization, 2014, 10 (2) : 503-519. doi: 10.3934/jimo.2014.10.503 [8] Jan-Hendrik Webert, Philip E. Gill, Sven-Joachim Kimmerle, Matthias Gerdts. A study of structure-exploiting SQP algorithms for an optimal control problem with coupled hyperbolic and ordinary differential equation constraints. Discrete and Continuous Dynamical Systems - S, 2018, 11 (6) : 1259-1282. doi: 10.3934/dcdss.2018071 [9] Alexander Tyatyushkin, Tatiana Zarodnyuk. Numerical method for solving optimal control problems with phase constraints. Numerical Algebra, Control and Optimization, 2017, 7 (4) : 481-492. doi: 10.3934/naco.2017030 [10] Theodore Tachim-Medjo. Optimal control of a two-phase flow model with state constraints. Mathematical Control and Related Fields, 2016, 6 (2) : 335-362. doi: 10.3934/mcrf.2016006 [11] Nidhal Gammoudi, Hasnaa Zidani. A differential game control problem with state constraints. Mathematical Control and Related Fields, 2022  doi: 10.3934/mcrf.2022008 [12] Ellina Grigorieva, Evgenii Khailov, Andrei Korobeinikov. An optimal control problem in HIV treatment. Conference Publications, 2013, 2013 (special) : 311-322. doi: 10.3934/proc.2013.2013.311 [13] Andrea Bacchiocchi, Germana Giombini. An optimal control problem of monetary policy. Discrete and Continuous Dynamical Systems - B, 2021, 26 (11) : 5769-5786. doi: 10.3934/dcdsb.2021224 [14] Anna Maria Candela, J.L. Flores, M. Sánchez. A quadratic Bolza-type problem in a non-complete Riemannian manifold. Conference Publications, 2003, 2003 (Special) : 173-181. doi: 10.3934/proc.2003.2003.173 [15] V.N. Malozemov, A.V. Omelchenko. On a discrete optimal control problem with an explicit solution. Journal of Industrial and Management Optimization, 2006, 2 (1) : 55-62. doi: 10.3934/jimo.2006.2.55 [16] Urszula Ledzewicz, Heinz Schättler. Drug resistance in cancer chemotherapy as an optimal control problem. Discrete and Continuous Dynamical Systems - B, 2006, 6 (1) : 129-150. doi: 10.3934/dcdsb.2006.6.129 [17] Lijuan Wang, Qishu Yan. Optimal control problem for exact synchronization of parabolic system. Mathematical Control and Related Fields, 2019, 9 (3) : 411-424. doi: 10.3934/mcrf.2019019 [18] Akram Kheirabadi, Asadollah Mahmoudzadeh Vaziri, Sohrab Effati. Solving optimal control problem using Hermite wavelet. Numerical Algebra, Control and Optimization, 2019, 9 (1) : 101-112. doi: 10.3934/naco.2019008 [19] Renzhao Chen, Xuezhang Hou. An optimal osmotic control problem for a concrete dam system. Communications on Pure and Applied Analysis, 2021, 20 (6) : 2341-2359. doi: 10.3934/cpaa.2021082 [20] M. Teresa T. Monteiro, Isabel Espírito Santo, Helena Sofia Rodrigues. An optimal control problem applied to a wastewater treatment plant. Discrete and Continuous Dynamical Systems - S, 2022, 15 (3) : 587-601. doi: 10.3934/dcdss.2021153

Impact Factor:

## Tools

Article outline

Figures and Tables