Detectability via observability in a nonuniform framework: dual relationship with controllability and stabilizability

The paper’s main claim (NUCO ⇒ NUED) is proved by chaining: (i) NUCO of (A,C) ⇔ NUCC of the dual (Theorem 6, with explicit time reversal and Gramian identities M ↔ K_d and N ↔ W_d) , (ii) a Riccati-based stabilizability theorem for NUCC systems (Theorem 10), which requires additional hypotheses (H1: nonuniform bounded growth; H2: specific NUCC constants; plus the existence of a Riccati solution) , and (iii) transfer of NUES from the dual to the original error dynamics via the dual-stability equivalence (Theorem 8) . However, the proof of Theorem 9 does not verify that the dual plant satisfies H1 or that the Riccati solution exists under NUCO alone, and Theorem 10 explicitly assumes both (H1) and the Riccati solution a priori, therefore the argument is not fully closed in the paper’s current form . The model’s solution follows the same high-level pathway but makes two key technical errors/omissions: it (a) identifies the dual without the paper’s required time reversal and conflates W and K-type Gramians, whereas the paper’s identities use time-reversed dual variables (3.12)–(3.13) ; and (b) transfers stability via the incorrect identity Φ_{A−K^T C}(t,s)^T = Φ_{A^T−C^T K}(t,s), instead of using the dual-time relation codified in Theorem 8 (which involves time reversal) . It also omits the extra hypotheses needed to invoke the nonuniform Riccati stabilizability step (H1/H2 and existence) . Hence, both the paper and the model are incomplete as written.