Unknown Input Observer Design for a class of Semilinear Hyperbolic Systems with Dynamic Boundary Conditions (Extended Version)

The candidate solution’s Task A reproduces the paper’s decoupling construction exactly: with R = I − HCM, K = K1 + K2, K2 = FH, and F = A − HCM A − K1CM, the error ODE becomes ε̇χ = F εχ + RE w − L N εx(t,1) and the error PDE keeps the transport form with the incremental nonlinearity ρ; this is precisely Proposition 1 and equations (5)–(6) in the paper. Choosing H = E((CME)⊤CME)^{-1}(CME)⊤ yields RE = (I−HCM)E = 0 under Assumption 1 (CME full column rank), which gives the fully decoupled error system (8) as in the paper’s derivation . For Task B, the candidate selects the same weighted energy functional V with e^{−μz} and derives its time derivative via integration by parts, adds the incremental sector bound through an S-procedure multiplier κ, and arrives at the same pointwise LMI D(z) ≺ 0 (cf. eq. (15)) together with the boundary/ODE LMI [ −e^{−μ}P −N⊤L⊤Q; • He(QF) + M⊤PM ] ≺ 0 (cf. eq. (14)); these are exactly the sufficient conditions in Theorem 1 guaranteeing UGES of the origin . The model also notes that the error dynamics are independent of the known input u after the chosen R, consistent with Gu = 0 under R = I−HCM in the paper’s setup . The paper is careful about well-posedness, invoking the incremental sector condition for global Lipschitz continuity (Lemma 1) and semigroup arguments for existence and uniqueness; while the candidate alludes to this, it does not dwell on it, which is acceptable for the posed tasks . Overall, both the paper and the model present the same Lyapunov-LMI proof structure for the detectable case, reaching identical LMIs and conclusions.