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

Mourad Azi; Mohand Ouamer Bibi

doi:10.3934/naco.2021005

Article Contents

2022, Volume 12, Issue 2: 279-291. Doi: 10.3934/naco.2021005

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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
^* Corresponding author: Mourad Azi

Received: May 2020

Revised: January 2021

Early access: February 2021

Published: June 2022

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Abstract

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.

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Mathematics Subject Classification: Primary: 49N05, 91G50; Secondary: 49M05.

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

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

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