Stability-Certified Learning of Control Systems with Quadratic Nonlinearities

The paper’s Theorem 1 proves BIBS stability for the quadratic control system ẋ = (J−R)x + H(x⊗x) + Bu when A = J−R with J = −J⊤, R ≻ 0 and H is energy-preserving, by using the quadratic energy E(x) = 1/2||x||^2 and showing Ė ≤ −σ_min(R)||x||^2 + ||B|| ||u||_∞ ||x||, leading to monotone decrease outside a radius r = ||B|| ||u||_∞ / σ_min(R) and the bound ||x(t)|| ≤ max{||x0||, r} (Theorem 1) . The candidate solution follows the same Lyapunov-energy route but strengthens the argument with a scalar differential inequality for ||x||, Dini-derivative handling at x = 0, forward invariance of the ball, and an explicit exponential estimate. Aside from minor typographical issues in the paper (e.g., mixing ||x|| and ||x||^2 around the threshold r), both arguments are correct and rely on the same core mechanism: skew-symmetry of J, positive definiteness of R, and energy preservation of H .