A perturbation analysis of parametric resonance and periodic control in spatially distributed systems

We consider a class of spatially distributed systems with spatially periodic coefficients that enter the system as perturbation. We then analyze the stability properties of the system for various frequencies of the perturbation. We show that, up to first order in the perturbation amplitude, parametric resonance can only occur at the isolated frequency equal to twice the frequency of the marginally-stable mode of the unperturbed system. Finally, we derive explicit expressions for the perturbation expansion of the marginally-stable mode, for all frequencies of the perturbation, and show how strict stability can be achieved for an appropriate choice of the perturbation.