Perturbation Methods in Stability and Norm Analysis of Spatially Periodic Systems

We consider systems governed by partial differential equations with spatially periodic coefficients over unbounded domains. These spatially periodic systems are considered as perturbations of spatially invariant ones, and we develop perturbation methods to study their stability and H system norm. The operator Lyapunov equations characterizing the H norm are studied using a special frequency representation, and formulae are given for the perturbation expansion of their solution. The structure of these equations allows for a recursive method for solving for the expansion terms. Our analysis provides conditions that capture possible resonances between the periodic coefficients and the spatially invariant part of the system. These conditions can be regarded as useful guidelines when spatially periodic coefficients are to be designed to increase/decrease the H norm of a spatially distributed system. The developed perturbation framework also gives simple conditions for checking exponential stability.