In this paper, the study of problems in optimal control is very important in our day life and their applications can be studied in many disciplines based on mathematical modeling physics, chemistry and economy. Because of the complexity of most applications, optimal control problems are solved numerically. New techniques for achieving an approximate solution to optimal control problems are considered. They are based upon B-spline polynomials approximation with state parameterization method. New useful property of B-spline polynomials is first derived then, it is utilized to propose a modified restarted technique to reduce the number of unknown parameters with fast convergence. Furthermore, it can be proved that with special knot sequence, the B-spline basis are exactly Bernstein polynomials. The objective of the present work is to propose an approximate technique for solving linear and nonlinear optimal control problems is presented. The algorithm modifies previous works to certain optimal control problems and is depended on a Bernstein series expansion of state parameterization. The differential expressions from the constraint and the cost index as well as the boundary conditions are reduced into algebraic equations. The technique starts from initial trajectory is based on the boundary conditions then new iterative method with the help Bernstein polynomials and produces satisfactory convergence with small number of unknown parameters. The applicability of the proposed algorithm is illustrated on four linear and nonlinear optimal control problems. The comparison with other works is also included in this paper.

Bernstein Polynomials (BEPs), Optimal Control Problems (OCPs), Parameterization Technique