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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

Mourad Azi 1,, and  Mohand Ouamer Bibi 2,

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

* Corresponding author: Mourad Azi

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.

Keywords: Optimal Control, Bolza Problem, Intermediate Phase Constraints, Support Problem, Cash Management.
Mathematics Subject Classification: Primary: 49N05, 91G50; Secondary: 49M05.
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.

Figure 1.  Optimal control $ u_1^*(t) $
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Figure 2.  Optimal control $ u_2^*(t) $
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