A Computational Method for H2-optimal Estimator and State Feedback Controller Synthesis for PDEs

The candidate solution reproduces the H2 upper-bound results for the estimator and the state-feedback controller under the LPI conditions that the paper states in Corollary 9 (Eq. 12) and Corollary 10 (Eq. 14). For the estimator, the model’s Lyapunov/storage-function derivation is essentially the same algebra underlying the paper’s use of Theorem 8 (Part 1) and a Schur complement. For the controller, the model proves the bound by lifting to p = T x, introducing T^{-1}, and using an observability Gramian; the paper instead uses the dual formulation (Theorem 8, Part 2) and a trace bound, avoiding assumptions about T^{-1} and Gramian existence. Hence both are valid; the model’s controller proof requires extra assumptions (invertibility of T and Gramian existence) not needed in the paper, but the core inequalities and conclusions match.