Estimation of the input-to-state stability gain functions from finite-dimensional approximations

Since the concept of input-to-state stability (ISS) was introduced, it has been extensively investigated for finite-dimensional control systems and has recently received attention for infinite-dimensional systems. While numerical techniques provide a bridge between these two worlds, a rigorous connection between the ISS of an infinite-dimensional system with an unbounded control operator and the properties of its finite-dimensional approximations has not yet been established. In this manuscript, we make a first step towards closing this gap by investigating numerical approximations of linear (boundary) control systems using semigroup theory. Specifically, we focus on linear boundary control systems where the autonomous evolution is governed by an analytic semigroup. For these systems, we show that ISS gains can be computed from approximations. We illustrate the applicability of these findings using a one-dimensional heat equation with Dirichlet boundary control for which reference ISS gains are known.