Extension of the Partial Integral Equation Representation to GPDE Input-Output Systems

The paper proves the GPDE↔PIE equivalence of internal exponential stability (Theorem 6.7) rigorously via an isometric PIE↔GPDE transformation T, norm equivalence on X0,0, and solution equivalence (Corollary 5.3). In contrast, the model’s proof sketch incorrectly sets the boundary variable v ≡ 0 under “zero input” and attempts to invert J[x; x̂] = [x; Dx̂] using only g = Dx̂, omitting the necessary Tv Cv x term required even with w = u = 0. This omission breaks the claimed bounded inverse J−1[x; g] = [x; Γg] in the general GPDE setting. While the model’s conclusion aligns with the paper, its proof is flawed by missing hypotheses and an incorrect inverse construction.