A new Lagrangian approach to optimal control of second-order systems

The paper’s Theorem 3.7 explicitly states the identities L̃E = H−1 ∘ αE_Q and H̃E = − H−1 ∘ αE_Q ∘ (βE_Q)−1 and indicates they can be verified in local adapted coordinates . The candidate solution does exactly that: it evaluates Pontryagin’s H−1 on αE_Q(q, κ, vq, vκ, u) to recover L̃E, and then uses the fiber derivative of L̃E and the energy identity to compute H̃E, finally matching it with −H−1 evaluated on αE_Q((βE_Q)−1(q, κ, pq, pκ, u)). All intermediate ingredients agree with the paper’s definitions: the new control Lagrangian L̃E(q, κ, vq, vκ, u) = vκ·vq + κ·Xv(q, vq, u) − C(q, vq, u) ; the extended Tulczyjew maps αE_Q and βE_Q in local coordinates (αE_Q: (q, κ, vq, vκ, u) ↦ (q, vq, vκ, κ, u), (βE_Q)−1: (q, κ, pq, pκ, u) ↦ (q, κ, vq = pκ, vκ = −pq, u)) ; the fiber derivative FL̃E with p⊤q = v⊤κ + κ⊤D2Xv − D2C and p⊤κ = v⊤q ; and the resulting new control Hamiltonian H̃E(q, κ, pq, pκ, u) = pq·pκ − κ·Xv(q, pκ, u) + C(q, pκ, u) . Pontryagin’s Hamiltonian for M = TQ is Hλ0(q, v, λq, λv, u) = ⟨(λq, λv), (v, Xv(q, v, u))⟩ + λ0C(q, v, u), so with λ0 = −1 we get precisely the form used by the candidate . The only tacit assumption the candidate uses is the invertibility of FL̃E, which the paper proves (L̃E is hyperregular) . Thus, both are correct and follow the same local-coordinate verification route.