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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 |
The aim of this work is to apply the results of R. Gabasov et al. [
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. Google Scholar |
[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. Google Scholar |
[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. |


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