  A small-gain approach is proposed to analyze closed-loop stability of linear
diffusion-reaction systems under finite-dimensional observer-based state
feedback control. For this, the decomposition of the infinite-dimensional
system into a finite-dimensional slow subsystem used for design and an
infinite-dimensional residual fast subsystem is considered. The effect of
observer spillover in terms of a particular (dynamic) interconnection of the
subsystems is thoroughly analyzed for in-domain and boundary control as well as
sensing. This leads to the application of a small-gain theorem for
interconnected systems based on input-to-output stability and unbounded
observability properties. Moreover, an approach is presented for the
computation of the required dimension of the slow subsystem used for controller
design. Simulation scenarios for both scalar and coupled linear
diffusion-reaction systems are used to underline the theoretical assessment and
to give insight into the resulting properties of the interconnected systems.
