Optimal Control of Nonlinear Systems Using the Shifted Legendre Polynomials

A numerical technique based on Legendre Polynomials for finding the optimal control of nonlinear systems with quadratic performance index is presented. An operational matrix of integration and product matrix are introduced and are used to reduce the nonlinear differential equations for the solution of nonlinear algebraic equations. The optimal solution from two classes of first and second order nonlinear systems is considered. In the case of second-order nonlinear systems, a new approach is introduced to find the optimal solution. In both cases, numerical examples are given and compared with the Taylor polynomials to confirm the accuracy of the proposed method.