Performance limitations for distributed systems via spatial-frequency Bode integrals

The Bode integral expresses a standard performance limitation for (almost) any controller that asymptotically stabilizes a linear time-invariant system. For the control of distributed systems, spatial invariance allows to write one such `Bode time-integral´ per spatial frequency. The present paper inverses the roles of spatial and temporal independent variables in this latter viewpoint. By transposing the notions of controller, causality, and asymptotic stability to the spatial variable, we obtain and interpret `Bode space-integrals´, one per temporal frequency. The result directly connects to the notion of string stability.