NUMERICAL ALGEBRA, doi:10.3934/naco.2021005 CONTROL AND OPTIMIZATION Volume 12, Number 2, June 2022 pp. 279–291 OPTIMAL CONTROL OF A DYNAMICAL SYSTEM WITH INTERMEDIATE PHASE CONSTRAINTS AND APPLICATIONS IN CASH MANAGEMENT Mourad Azi ∗ Department of Mathematics and Computer Science University of Mila, 043000 Mila, Algeria Research Unit LaMOS, University of Bejaia Mohand Ouamer Bibi Research Unit LaMOS, Department of Operational Research University of Bejaia, 06000 Bejaia, Algeria (Communicated by Chongyang Liu) 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 de- veloped 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 over- drafts 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. Introduction. Optimal control theory is an important area of applied mathemat- ics, developed to find optimal way to control management systems, overcome the arduous tasks, predict and control future events and finally to optimize a certain criterion. This theory is applied in many fields of sciences, in particular engineering, physics, biology, economics and finance [24, 26, 30, 32]. Optimal control with intermediate phase constraints is often used to solve prob- lems in which state constraints are imposed on intermediate times. On the other hand, these problems arise as auxiliary tasks for solving a nonlinear optimal control problem without phase constraints [21]. The optimal control problems in Bolza 2020 Mathematics Subject Classification. Primary: 49N05, 91G50; Secondary: 49M05. Key words and phrases. Optimal Control, Bolza Problem, Intermediate Phase Constraints, Support Problem, Cash Management. * Corresponding author: Mourad Azi. 279