The maximum principle without the adjoint equation

The Classical Pontryagin maximum principle for boundary trajectories of control systems consists of two parts: the maximum condition and the adjoint equation. In this paper, we study the maximum condition and minimal assumptions under which it holds for boundary trajectories of differential inclusions. Not surprisingly, the maximum condition alone is satisfied under much weaker assumptions than the full maximum principle. We prove it under mild hypotheses of measurability and lower continuity. In particular, we allow inclusions with nonconvex, unbounded values, not continuous in the state variable. © 1989