Modified Pell Matrix Technique for Solving Optimal Control Problems

The orthogonal polynomial basis functions are used to solve different mathematical problems, especially for optimal control and many other engineering problems, which attract many researchers to work on. In this study, the modified Pell polynomials (MPPs) are presented and their new properties are investigated to be used for solution approximation of optimal control problems. Some formulas for MPPs are derived by matrices. A new exact formula expressing the derivatives of MPPs explicitly of any degree is constructed. The main advantage of the presented formulas is that the new properties of MPPs greatly simplify the original problems and the result will lead to easy calculation of the coefficients of expansion, it also increases the accuracy and reduces the computational time. A new computational method along with the MPPs is proposed to solve one of the optimal control problems. Numerical results are included to demonstrate the validity of this new technique. It shows an important improvement in error approximation when the polynomial degree is increased.