This work is devoted to find the critical time instants for a multivariate quantum harmonic oscillator system under optimal control. All quantum optimal control agents are taken linear in their arguments. The objective operator and the operator whose expectation value is to be suppressed are expressed as linear combinations of the position and momentum operators. The external field influence on the system is approximated only by the dipole polarizability term and the dipole function is taken as a linear combination of the positions. These linearities together with the elastic force structure of the mutual interactions amongst the oscillators brings the whole linearity to the system. We do not concern with the wave and costate functions. Instead, the expectation values of the position and momenta are taken as unknowns together with the two way transitions between the states described by the wave and costate functions, through positions and momenta. The resulting control problem appears to be a boundary value problem in time through a vector valued ODE. The critical time durations of this equation are sought by expanding the deviation parameter with respect to a perturbation parameter. Perturbation terms for these critical time instants can be evaluated analytically.