Weak solutions of port-Hamiltonian systems

The PDF’s Theorem 3.2 asserts that, under the weak Dirac testing identity (3.5) and assuming the occurring co-energy space E (norm (3.2)) is reflexive, the canonical projection ιx belongs to W^{1,p}_loc(I;E′). The paper proves this by: (i) testing with time-scaled witnesses from D to derive the scalar identity (3.9); (ii) organizing witnesses via D0, D1 and a bounded bilinear form ⟨⟨⟨·,·⟩⟩⟩ (3.6)–(3.8); and (iii) invoking reflexivity/RNP and a Sobolev–Reshetnyak criterion to conclude ιx ∈ W^{1,p}_loc(I;E′) . The candidate solution follows the same initial scalarization (test along fixed e0∈E using δ(t)-scaled witnesses) to get a scalar W^{1,p}_loc derivative and independence of witnesses—mirroring the paper’s use of D0—but then establishes an absolute continuity estimate ∥ιx(t)−ιx(s)∥ ≤ ∫ L(τ)dτ and uses the Radon–Nikodým property to pass to the Bochner–Sobolev conclusion. Thus, both are correct; the paper uses the D1/Reshetnyak route, while the model uses an AC/RNP route. The only minor divergence is that the paper explicitly isolates a constant m in the derivative bound via the bounded bilinear form, whereas the model implicitly sets m=1 through the choice of norms; this is a harmless normalization difference. Overall, the arguments agree on hypotheses, steps, and conclusion .