A note on robustness of linear spatially distributed parameter systems and their numerical approximations

In this paper, we investigate a relationship between robust stability properties of linear spatially distributed parameter systems (LSDPS) with disturbances and robust stability properties of their numerical approximations. Since it is hard to analytically find solutions of a partial differential equation, numerical methods, such as finite-difference methods, are always used to approximately find the solutions. Moreover, it is crucial that the numerical method reproduces (approximately) the behavior of the actual system model. For instance, if the actual system is stable in some sense, then the numerical method should possess (approximately) the same stability property and vice versa. Our results show that input-to-state exponential stability (ISES) properties of the numerical approximation with respect to disturbances are equivalent to practical ISES of the LSDPS provided that: (i) the finite-difference approximation is consistent with the model; (ii) an appropriate uniform boundedness condition holds for the numerical method. Our results can be regarded as an extension of the celebrated Lax- Richtmyer theorem to systems with disturbances, as well as its application to analysis of ISES. This question is typically not considered in the numerical analysis literature and yet it is very well noticed by in control applications.