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 H² system norm. The operator Lyapunov equations characterizing the H² norm are studied by using a special frequency representation, and formulas are given for the perturbation expansion of their solution. The structure of these equations allows for a recursive method of 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 or decrease the H² norm of a spatially distributed system. The developed perturbation framework also gives simple conditions for checking whether a spatially periodic operator generates a holomorphic C₀ semigroup and thus satisfies the spectrum-determined growth condition.