OPTIMAL TIME AVERAGES IN NON-AUTONOMOUS NONLINEAR DYNAMICAL SYSTEMS

The paper explicitly establishes the auxiliary-function variational equality Φ* = inf_{V∈C^1(B)} max_{x∈B} [Φ(x) + f(x)·∇V(x)] (with a sketch proof via invariant measures and a minimax swap), and builds non-autonomous/trigonometric ODEs into autonomous polynomial systems using the 2D oscillator trick; see the equality (1.9) and its sketch, and the constructions in (4.1)–(4.8) . The candidate solution’s Parts A–C mirror these arguments: (A) the Hopf-type oscillator autonomization and restriction to the invariant circle S^1; (B) the treatment of a variable entering only via sin/cos using a state-dependent angular speed; and (C) the convex dual identity proved via time averages and invariant measures. Minor differences are present but benign: the paper discusses excluding the spurious y≡0 oscillator solution via the S-procedure or penalty terms, whereas the model restricts to B×S^1; and in case (B) the paper notes Φ should not depend on the eliminated angle except through cos/sin, or one can evolve it passively, which the model essentially does. Overall, the proofs align and are correct within the paper’s assumptions.