This paper deals with synchronization of a class of infinite-dimensional systems. The considered network is described by a collection of semilinear Lipschitz boundary-actuated infinite-dimensional dynamics. For undirected connected graphs, sufficient conditions for asymptotic synchronization are established. We show that the proposed conditions when applied to systems of hyperbolic semilinear conservation laws can be recast into a set of matrix inequalities. For this class of systems, sufficient conditions in the form of linear matrix inequalities for the design of synchronizing policies are provided.