 A sufficient condition for a maximum is the concavity of the Hamiltonian evaluated at the solution, i.e.where formula_17 is the optimal control, and formula_18 is resulting optimal trajectory for the state variable.Alternatively, by a result due to Olvi L. Mangasarian, the necessary conditions are sufficient if the functions formula_14 and formula_34 are both concave in formula_10 and formula_6.