H∞-optimal control of coupled ODE-PDE systems using PIE framework and LPIs

The paper proves the H-infinity (bounded-real) result for PIE systems via a dual Lyapunov/dissipation argument that avoids inverting the operator T and uses an auxiliary slack variable v to complete the square; see Theorem 17 and its proof steps yielding V̇ − γ||w||^2 − γ||v||^2 + 2⟨v,z⟩ ≤ 0 and the choice v = z/γ, which gives the L2-gain bound without assuming T is invertible . The model solution, by contrast, (i) assumes T is boundedly invertible and rewrites ẋ = T^{-1}(A+B_2K)x + T^{-1}B_1w, a hypothesis not required (and often deliberately avoided) in the paper, and (ii) claims a direct “spatial KYP identity” d/dt⟨x,Px⟩ = ⟨x,Sym_T(AP+B_2Z)x⟩ + … without the dual-system derivation used in the paper. It also identifies the 3×3 4‑PI LPI with a dissipation inequality in the variables (z,w,x) instead of the paper’s (w,v,x), thereby skipping the essential completion-of-squares step with v = z/γ. These gaps make the model’s derivation incorrect under the paper’s stated hypotheses. The paper’s argument is internally consistent and complete, including the dual L2-gain equivalence and its use in the controller proof .