Parametric resonance in spatially distributed systems

We consider spatially distributed systems described by partial differential equations (PDEs) in which some of the coefficients are spatially periodic functions. Such systems arise in certain distributed sensorless control schemes which we term spatio-temporal vibrational control, which is a generalized version of standard temporal vibrational control. The mechanism by which certain sensorless periodic feedbacks stabilize or destabilize systems is more generally known as parametric resonance. We develop a spatio-temporal lifting framework using which we analyze stability and system norms of PDEs with periodic coefficients. Examples of PDEs in which parametric resonance occurs are given.