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In chapter one: a general literature review is presented, including an overview of the optimal control theory with the basic definitions associated with it. In chapter two: some fundamental concepts concerning orthogonal functions are given. Some new important properties and formulas of shifted Legendre polynomials, and functions are derived. Derivations of such formulas are essential in our work in order to simplify the computation. In chapter three: a method of approximating the QOCP is described to convert the optimal control problem into a standard quadratic problem using the state vector parameterization technique. Furthermore, computational results for some examples are included for illustration the proposed method. conclusions and further research suggestions are included in chapter four.