Conditions for l∞ and l2 system robustness for spatially invariant systems

We consider spatiotemporal systems and study their l_infin and I₂ robustness properties in the presence of spatiotemporal perturbations. In particular, we consider spatially invariant nominal models and provide necessary and sufficient conditions for system robustness when the underlying perturbations are linear time and spatially varying, unstructured and structured. It turns out that these conditions are analogous to the scaled small gain condition (which is equivalent to a spectral radius condition and an LMI for the l_infin and I₂ case respectively) derived for standard linear time invariant models subject to linear time varying perturbations.