New adapted spectral method for solving stochastic optimal control problem

Document Type : Research Paper

Authors

Mathematical Analysis and Applications Laboratory, Departement of Mathematics, Faculty of Mathematics and Informatics, Mohamed El Bachir El Ibrahimi university of Bordj Bou Arreridj, El Anasser, 34030, Algeria

10.22075/ijnaa.2023.30665.4464

Abstract

Optimal control theory is a branch of mathematics. It is developed to find optimal ways to control a dynamic system. In 1957, R.Bellman applied dynamic programming to solve optimal control of discrete-time systems. His procedure resulted in closed-loop,  generally nonlinear, and feedback schemes. Optimal control problems which will be tackled involve the minimization of a cost function subject to constraints on the state vector and the control. Lagrange multipliers provide a method of converting a constrained minimization problem into an unconstrained minimization problem of higher order. The necessary condition for optimality can be obtained as the solution of the unconstrained optimization problem of the Lagrange function and the bordered Hessian matrix is used for the second-derivative test. A spectral method for solving optimal control problems is presented. The method is based on Bernoulli polynomials approximation. By using the Bernoulli operational matrix of integration and the Lagrangian function, stochastic optimal control is transformed into an optimisation problem, where the unknown Bernoulli coefficients are determined by using Newton's iterative method. The convergence analysis of the proposed method is given. The simulation results based on the Monte-Carlo technique prove the performance of the proposed method. Some error estimations are provided and illustrative examples are also included to demonstrate the efficiency and applicability of the proposed method.

Keywords

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Articles in Press, Corrected Proof
Available Online from 10 December 2023
  • Receive Date: 16 May 2023
  • Revise Date: 28 September 2023
  • Accept Date: 13 October 2023