Consider the following problem: \begin{align} &\max_{u(\cdot)}\int_0^\infty{e^{-rt}F(x(t),u(t))dt}\\ \mathrm{s.t.}&\quad \dot x(t) = f(x(t),u(t)), x(0)= x_0. \end{align} Let $H = F + \lambda f$ denote the Hamiltonian. Pontryagin's maximum principle provides necessary optimality conditions \begin{align} &0 = \frac{\partial H}{\partial u},\\[2mm] &\dot \lambda = r \lambda - \frac{\partial H}{\partial x},\\[2mm] &\lim_{t \to \infty}e^{-rt}\lambda(t)x(t) = 0. \end{align} For sufficiency one usually requires that $H$ is concave in $(x,u)$ [Mangasarian] or that the maximized Hamiltonian is concave in $x$ [Arrow]. I read, however, the following sentence in a paper (pdf):
In fact sufficiency is satisfied for trajectories converging to the locally stable steady states of the cooperative solution and open-loop equilibria.
I've never seen this and was wondering whether there exists a formal proof for this claim.