Damping Feedback Stabilization for Cyclic Interconnections Systems: Oscillations Suppression and Synchronization

This paper considers the problem of stabilizing (bio)chemical reaction networks that can be represented as cyclic interconnections. The objective of the paper is to present a constructive way to compute a dissipative potential function for the system. This potential is then used for constructing a smooth feedback stabilizing controller. We obtain a characteristic one-form for the system by taking the interior product of a non vanishing two-form with respect to the drift vector field. A homotopy operator is then constructed locally around the desired equilibrium, leading to the computation of a dissipative potential for the system. The dynamics of the system is then decomposed into an exact part and an anti-exact one. The exact part is generated by a potential, that is used to construct the smooth stabilizing feedback, under the so-called weak Jurdjevic-Quinn conditions. We consider the problems of oscillations suppression and synchronization of oscillators as illustrations of potential applications of the proposed method.