Leader-follower synchronization of a network of boundary-controlled parabolic equations with in-domain coupling

In this paper, we study the leader-synchronization problem for a class of partial differential equations with boundary control and in-domain coupling. We describe the problem in an abstract formulation and we specialize it to a network of parabolic partial differential equations. We consider a setting in which a subset of the followers is connected to the leader through a boundary control, while interconnections among the followers are enforced by distributed in-domain couplings. Sufficient conditions in the form of matrix inequalities for the selection of the control parameters enforcing exponential synchronization are given. Numerical simulations illustrate and corroborate the theoretical findings.