Pontryagin's Maximum Principle as a sufficient condition? It is know that Pontryagin's maximum principle provides in general only a necessary condition in the following sense: The ODE system which is known to be solved by the optimal control may have multiple solutions in general. Thus, solving the ODE system does not necessarily provide one with the optimal control.
I am not an expert in control theory and optimization but after having a go with the literature it seems to be the case that sufficiency can be guaranteed only in a few nonlinear cases. 
Here my question: Does anybody know whether the maximum principle is a sufficient or necessary condition if the ODE system to be controlled is given in terms of polynomials (in several variables)?
 A: In the Book by Piccoli and Bressan, Introduction to Mathematical Control Theory, you can find your answer in the chapter 7 of sufficient condition for optimality. Basically you have:


*

*PMP + unique admissible control. If there exists only an admissible control that satisfy PMP, that control is optimal.

*PMP + convexity, if your functional is convex w.r.t. the control and the set of admissible controls is also convex, the trajectories that verify PMP are optimal.

*If all the solutions of PMP for initial conditions covers the phase space in a particular way you have sufficiency.


The first point is obvious. For a deeper explanation of the third point is better to directly refer to literature. For the second I only want to remember that convexity must be understood w.r.t. a set of functions, generally subset of $L^\infty$, beacuse the value of the functional depends on the trajectory of the system under the action of the control law.
Your particular where the "ODE system to be controlled is given in terms of polynomials" is rather obscure. I figure something like
$$
\dot x^i = A^i + B^{i}_j x^j + C^{i}_{jk} x^j u^k + D^i_j u^j +
E^i_{jk} x^j x^k + F^i_{jkl} x^j x^k u^l + G^i_{ij} u^j u^k + \cdots
$$
So your description is rather general.
