CarboNet: A Finite-Time Combustion-Tolerant Compartmental Network for Tropospheric Carbon Control

Full-state LQR: the paper’s matrices, weights, and optimal gain match the model’s computation (A in eq. (39), B in eq. (9), R1 = I4, r2 = 1; K ≈ [-0.94, -0.67, -0.93, -0.65]) . However, the paper’s reported “x1 reaches x1e in ≈25 days” (full-state) and “≈60 days” (output feedback) are informal, plot-based times without a precise event definition; the model, by contrast, computes the first crossing time (≈0.389 d) and observes subsequent undershoot/return, which explains the discrepancy without falsifying either given the ambiguous notion of ‘reach’ . Output feedback: the paper solves the Ilka–Murgovski OF-LQR and reports K = -0.837 and u(0) ≈ 260 t/d; the model instead picks K = -0.2 (stable and admissible) and solves the associated Lyapunov equation equivalent to the OF-LQR condition, but does not optimize K; thus it is correct but not the same task as the paper’s optimized OF-LQR . Finally, the paper’s statement that x2 and x3 are “brought to zero in ≈6 days” is mathematically inaccurate for stable continuous-time exponentials (they never reach zero in finite time); a threshold must be specified. The model correctly flags this and reports finite-time thresholds instead . The net-zero identity d/dt(x1+x4) = ϕnz(t) and the implication ϕnz ≡ 0 ⇒ x1+x4 = const are consistent between paper and model .