Solving optimal control problems using the Picard’s Iteration Method Abderrahmane Akkouche a, b∗ , and Mohamed Aidene b a D´ epartement de Math´ ematiques, Facult´ e des Sciences et des Sciences Appliqu´ ees, Universit´ e AKLI MOHAND OULHADJ de Bouira, 10 000 Bouira, Alg´ erie. b Laboratoire de Conception et Conduite des Syst` emes de Production Universit´ e MOULOUD MAMMERI de Tizi-Ouzou, 15 000 Tizi-Ouzou, Alg´ erie. May 29, 2019 Abstract In this paper, the Picard’s iteration method is proposed to obtain an approximate analytical solution for linear and nonlinear optimal control problems with quadratic objective functional. It consists in deriving the necessary optimality conditions using the minimum principle of Pontryagin, which result in a two-point-boundary-value-problem (TPBVP). By applying the Picard’s iteration method to the resulting TPBVP, the optimal control law and the optimal trajectory are obtained in the form of a truncated series. The efficiency of the proposed technique for handling optimal control problems is illustrated by four numerical examples, and comparison with other methods is made. keywords :Optimal control, Pontryagin’s minimum principle, Hamilton-Pontryagin equa- tions, Picard’s iteration method, Ordinary differential equations. 1 Introduction Optimal control problems can be solved by direct or indirect methods [37]. Direct methods consist in converting the optimal control problem into an optimization one by discreetizing the state and the control variables, then the optimal control law is achieved using optimization methods [6]. However, the indirect methods consist in solving the necessary optimality conditions obtained from the application of the Pontryagin’s minimum principle [31]. This necessary optimality conditions are given by a set of first order ordinary differential equations, known as the Hamilton-Pontryagin equations, with appropriate boundary conditions that define a two-point-boundary-value-problem (TPBVP) [4]. In which, the optimal control law is determined by solving the resulting TPBVP using the shooting methods [21, 39]. But the shooting methods suffer from difficulties in finding an approximate initial guess for the unspecified conditions at one end that produce solution reasonably close to the specified condition at the other end, because the solution is often very sensitive to small changes in the unspecified boundary conditions [6, 5]. These numerical difficulties are augmented by the relatively small domain of convergence of the Newton method which is built in the shooting methods [7]. In the last decade, a variety of semi-analytic methods to solve linear and nonlinear ordinary differ- ential equations are presented [32]. These methods use practical iterative formulas to determine the solution or the approximate one of the problem in the form of an inifinite series that converges to the * Author to whom the correspondence should be addressed : akkouche.abdo@yahoo.fr 1 This provisional PDF is the accepted version. The article should be cited as: RAIRO: RO, doi: 10.1051/ro/2019057