The Pontryagin and Euler-Lagrange Necessary Conditions

For optimization problems three well developed theories of necessary conditions, named after Hamiltonian, Euler/Lagrange, and Pontryagin, exist. In a recent paper, Loewen and Rockafellar proved the existence of an adjoint arc simultaneously satisfying the Hamiltonian and Euler/Lagrange necessary conditions. We prove a similar result showing that there exists an adjoint arc obeying the Pontryagin and Euler/Lagrange necessary conditions