A direct method for the solution of optimal control problems of linear time-invariant systems with integral quadratic energy constraint is developed with the help of functional analysis. It is based on the representation of the state transition matrix of the adjoint system in terms of an orthonormal set of functions in a finite dimensional sub-space of square integrable Hilbert space and the Fourier expansion of the control vector in the Hilbert space. It is shown that the optimal control vector exists uniquely and is expressed as a linear functional of the orthonormal basis functions of the finite sub-space. The simplicity of the method lies ultimately in reducing the optimal problem to that of solving a finite set of linear algebraic equations