Pontryagin principle: does the abnormal multiplier define a minimum 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).
 A: Hello user 47662  (I feel talking with a robot ;)). You know that I have the same question, but I have just read this paper: Leonard D Berkovitz. Variational methods in problems of control and programming. Journal of Mathematical Analysis and Applications. In that paper appears a derivation of the PMP (Pontryagin Maximum Principle) from the calculus of variation. It is a good reading. There appear the PMP as a form of the Weiertrass necessary condition of convexity. I will speak only for systems that are non constrained in the space variable. If you take the control $u \in \mathcal{U} \subset \mathbb{R}^m$ to be (as the PMP says)
$
u=\arg\max_{u \in \mathcal{U}}  H(t,x,u,\lambda_0,\lambda)
$
you must guaranty the convexity $H_{uu} < 0$, i.e., the matrix with components $\frac{\partial^2 H}{\partial u_i \partial u_j}$ is negative definite. 
As far as I have understood, the PMP contains itself the Weierstrass and the Lagendre-Clebsh condition, so it must be a sufficient condition. In that paper, the condition about the multiplier $\lambda_0$ is not clear to me. You can also read the book by Bryson, "Applied Optimal Control", in the chapter 6, section 9, there are enumerated four necessary conditions


*

*Euler-Lagrange equations

*Legendre-Clebsh condition

*Weierstrass condition

*Non existence of conjugate points.


Weierstrass conditions implies Legendre-Clebsh conditions. I don't know what Bryson means with conjugate points it could be something about the multipliers $\lambda$ and $\lambda_0$. I hope I've been helpful.
