Construction of Lyapunov functions for interconnected parabolic systems: an iISS approach

This paper is devoted to two issues. One is to provide Lyapunov-based tools to establish integral input-to-state stability (iISS) and input-to-state stability (ISS) for some classes of nonlinear parabolic equations. The other is to provide a stability criterion for interconnections of iISS parabolic systems. The results addressing the former problem allow us to overcome obstacles arising in tackling the latter one. The results for the latter problem are a small-gain condition and a formula of Lyapunov functions which can be constructed for interconnections whenever the small-gain condition holds. It is demonstrated that for interconnections of partial differential equations, the choice of a right state and input spaces is crucial, in particular for iISS subsystems which are not ISS. As illustrative examples, stability of two highly nonlinear reaction-diffusion systems is established by the the proposed small-gain criterion.