Abstarct: In this thesis stability of discrete-time linear systems with output feedback via two special values and parametric nonlinear inverse matrix evaluated. Implementation of discrete-time fractional operators in robotics engineering and other branches of industry, especially electrical and mechanical engineering is very important. A key step in the implementation of the fractional digital transfer function, or discrete numerical calculation of fractional differentiation. There are many methods for discrete accordance with the type of system will be included. Reverse particular value problems in several disciplines such as design control, geophysics, circuit theory and applications. This is one of the most important applications, especially in the amounts allocated to it in the context of control theory. In this thesis a new method to solve the allocation problem eigenvalues of matrix inversion using eigenvalues and parametric nonlinear vector is baxsed companion form And then gaining feedback matrix switcher output deficit for the first time using this method was introduced. In the end, the examples listed for the effectiveness of using MATLAB software we offer. ‎‎ ‎Later in the thesis, we present a computational method for solving optimal control problems and the controlled Duffng oscillator. This method is baxsed on state parametrization. In fact, the state variable is approximated by Boubaker polynomials with unknown coeffcients. The equation of motion, performance index and boundary conditions are converted into some algebraic equations. Thus,an optimal control problem converts to a optimization problem,which can then be solved easily. By this method,the numerical value of the performance index is obtained. Also,the control and state variables can be approximated as functions of time. Convergence of the algorithms is proved. Numerical results are given for several test examples to demonstrate the applicability and effciency of the method.