Recent years brought a new angle of view to the problems involving matrix representations of, especially, functions. It is the utilization of the fluctuation free matrix representation of those functions, where the matrix representation of the function is taken as the image of the matrix representation of the independent variable under the concerned function as a first approximation. The quality of the approximation becomes better as the fluctuations are suppressed by changing the subspace dimension under consideration. In this study, we discuss the quantum optimal control problem of simple harmonic oscillator as a first step towards the construction of possibly an efficient new tool. Here, the expectation value based optimal control equations are formulated and the all agents of the problem are taken linear for simplicity. The resultant linear system of ordinary differantial equations have exact solutions. By using these solutions and the fluctuationlessness approximation, it is possible to evaluate the matrix representation of an analytic function of the position and momentum operators.