On l∞ stability and performance of spatiotemporally varying systems

A characterization of stability for spatiotemporally varying systems based on input-output description of the plant and controller is presented. The overall controller is generated by a collection of controllers indexed in space and time, which are based on frozen spatially and temporally invariant descriptions of the plant. In particular, we consider the case where these frozen controllers are not necessarily adjusted for every instance in space and time, and hence are used for some fixed window in time and space. It is shown that the actual spatiotemporally varying system can be stabilized using frozen in space and time controllers, provided the variations in the spatiotemporal dynamics are small enough. We also show that the l∞ to l∞ performance of such systems cannot be much worse than that of the frozen spatially and temporally invariant systems.