A behavioural approach to port-controlled systems

The paper constructs three continuous machines B0 (closed dissipative Hamiltonian), B (enclosing time-varying Hamiltonian interface), and B̂ (input–state–output pH system), together with monomorphisms A: B0 → B and Ξ: B̂ → B, and shows the interface diagram commutes, yielding that (B̂, Â, Ê) is a port-controlled B0-system with time-varying Hamiltonian interface. The candidate solution reproduces the same construction and checks the same properties (sheafhood, injectivity/monomorphisms, commuting A/E-projections), adding minor regularity checks and an (optional) residual E-map to clarify a small typing inconsistency around the constant zero sheaf O. The core argument and result align with Section 4.1 and Def. 3.5 of the paper, and the embeddings and projections match those given in the text. See the paper’s definitions of B0 and B with Hα, the embedding A, the projections ϕA and ϕE, and the embedding Ξ culminating in the stated port-controlled structure , as well as the interface notion in Def. 3.5 .