Abstarct: Solving infinite horizon Optimal control problems,by high accuracy numerical methods,has been considered by many researchers, since the exact solution of such problems is usually impossible. Hence, in the section 2 we present a pseudospectral method to solve infinite horizon optimal control problems. Here, we discuss the domain transformation technique for converting the infinite horizon problem into a finite horizon problem. A class of bounded pseudo spectral method is baxsed on the Gauss-Radu points and Gauss-Radu pseudo spectral method is extended for solving infinite horizon Optimal control problems. For this goal, by costate vectors and maximum principle, the optimal solution is approximated. Also,in the section 3 a new approach is given for designing the stabilizing control and adaptive control. We suggest an infinite horizon optimal control problem according to the nonlinear control system. we prove that the optimal control of this problem can be a stabilizer control or adaptive control.Then we convert the infinite horizon optimal control problem into an equivalent finite horizon optimal control problem. Moreover, we use the Legendre pseudo spectral method to solve the suggested problem. Finally, in the section 4, we present some optimality conditions for infinite horizon optimal control problems.