The Pontryagin principle PM provides the necessary condition for a local minimum of the functional
$ J(u)=\int L(x(t),u(t))dt \\$
subject to:
$\dot x = f(x(t),u(t)) \ \ \ \ x(t0)=x0, \ \ x(t1)=x1$,
Do I need to compute second order conditions to identify a minimum over a maximum, as we would do in calculus of variations?
It looks to me the the PM already excludes maxima by requiring the abnormal parameter $\lambda _0$ to be non-negative.$\lambda _0$ appears in the PM Hamiltonian
$H(x,u,\lambda_0,\lambda) = \lambda_0L(x,u)+\lambda ^Tf(x,u)$
but not in the calculus of variation. In the PM proof, $\lambda_0$ is used to ensure the terminal cone points "upward". If my understanding is correct, the principle then would provide sufficient conditions - and not only necessary - for a local minimum. Of course, these conditions would be only necessary conditions for global minimum (HJB provides the sufficient conditions for a global minimum).