Model-free Continuation of Periodic Orbits in Certain Nonlinear Systems Using Continuous-Time Adaptive Control

The paper derives the same error model ė = A e + b θ̃^T Q(t,q), uses the composite Lyapunov V = e^T P e + (1/Γ) θ̃^T θ̃ to obtain V̇ = −e^T S e ≤ 0, proves boundedness and e ∈ L2, then applies a Barbalat-type argument to conclude e(t) → 0 and θ̃̇ → 0. Under persistent excitation of Q(t,q), the paper shows θ̃(t) → 0 via a mean-value/PE argument and then that x and u converge to T-periodic steady states determined by g, with non-invasiveness when r = q*. These are precisely the statements and steps in the candidate solution, which follows the same structure (Lyapunov + Barbalat + PE) and arrives at the same conclusions. Minor presentational differences exist (the model invokes a standard MRAC PE theorem; the paper uses an integral perturbation argument), but there is no substantive conflict. See the paper’s equations (9)–(14) for the setup and Lyapunov estimate and (18)–(21) for the PE-based parameter convergence and periodic steady-state claims .