Existence of optimal controls for some nonlinear processes

We prove the existence of optimal controls for an ordinary differential system which is nonlinear in the state functionx but is linear in the control functionu, that is, $$dx/dt = f_1 (t,x) + f_2 (t,x)u$$ Rather weak regularity assumptions are made on the right-hand side of the above system, the constraints, and the cost functional.