This work focuses on the optimal control of a quantum system composed of harmonic oscillators under linear control agents (dipole function, objective operator, and the penalty operator whose expectation value is to be suppressed). The main purpose of the work is to determine the temporal external field amplitude function. Paper recalls the formulation of the optimal control equations first. Then a set of ordinary differential equations over the expectation values of certain unknown entities is constructed. These temporal differential equations have time varying coefficients unless the weight functions appearing in the cost functional are constant. Certain accompanying conditions are needed to get unique solutions. Investigations show that one half of the conditions should be given at the initial instant and the other half should be specified at the final moment. Since the differential equations contain another unknown entity, deviation parameter, solutions must satisfy an algebraic equation derived from the definition of this parameter. Results do not involve the explicit structure of the wave function and costate function. Only the external field amplitude and the deviation parameter are determined here. The evaluation of the wave function and costate function needs additional treatments to the control equations. We report certain analytical results for external field amplitude and the deviation parameter and give certain illustrative implementations to finalize the paper.