Existence of optimal control by using maximum principle

Relaxed control takes a crucial rule in existence theory of optimal control. In recent years, we used maximum principle that an optimal relaxed control satisfies to study the existence of optimal control, getting some sufficient conditions. Lately, we studied existence problem of optimal control for cost functional essentially having quadric form, getting some interesting results. In particular, we got some non-trivial examples of optimal control problems that admit no optimal control. In this paper, we will recall these results briefly.