Abstract The main result in this short note is that the integral form of the Leitmann–Stalford sufficiency conditions can be verified for a class of optimal control problems whose Hamiltonian is not concave with respect to the state variable. The main requirement for this class of problems is that the dynamics is sufficiently dissipative. An application to a Stackelberg differential game between a producer and a developer is exemplified. Using our result we show that the necessary conditions implied by Pontryagin’s maximum principle are also sufficient. This allows a complete characterization of the solution.