On l∞ and l₂ robustness of spatially invariant systems

We consider spatiotemporal systems and study their l∞ and l ₂ 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 for the cases when the underlying perturbations are linear spatiotemporal varying, and nonlinear spatiotemporal invariant, unstructured or structured. It turns out that these conditions are analogous to the scaled small gain condition (which is equivalent to a spectral radius condition and a linear matrix inequality for the l∞ and l₂ cases, respectively) derived for standard linear time-invariant models subject to time-varying linear and time-invariant nonlinear perturbations.