Consensus of marginally stable systems of reaction–diffusion equations with decentralized boundary control

This paper deals with the consensus problem for a network of identical reaction–diffusion equations with boundary control. Under the assumptions of open-loop marginal stability and connectivity of the interconnection graph, a decentralized control law can be designed such that the states of the equations converge to the same constant value. The synthesis of the controllers is carried out based on spectral decomposition, which is possible thanks to a clever input-dependent change of coordinates. Furthermore, the proposed architecture can be enhanced to cope with in-domain couplings among the networked equations. Two simulation examples support and illustrate the theoretical findings.