  We present a general approach to prove existence of solutions for optimal
control problems not based on typical convexity conditions which quite often
are very hard, if not impossible, to check. By taking advantage of several
relaxations of the problem, we isolate an assumption which guarantees the
existence of solutions of the original optimal control problem. To show the
validity of this crucial hypothesis through various means and in various
contexts is the main body of this contribution. In each such situation, we end
up with some existence result. In particular, we would like to stress a general
result that takes advantage of the particular structure of both the cost
functional and the state equation. One main motivation for our work here comes
from a model for guidance and control of ocean vehicles. Some explicit
existence results and comparison examples are given.
