On the Stability of Spatially Distributed Systems

In this note, a sufficient condition is derived for the stability of a spatially invariant distributed dynamical system, based on the geometrical structure of the null space of a matrix polynomial. This condition is less conservative than the available computationally feasible criteria. Moreover, using the idea of parameter dependent linear matrix inequalities, a necessary and sufficient condition is obtained. Both of these two conditions are expressed by LMIs. While the necessity of the latter is lost if the degree of the related matrix polynomials is small, its conservativeness can be sequentially reduced.