Summary: It is well known that every strong local minimizer of the Bolza problem under state constraints satisfies a constrained maximum principle. In the absence of constraints qualifications the maximum principle may be abnormal, that is, not involving the cost functions. Normality of the maximum principle can be investigated by studying reachable sets of an associated linear system under linearized state constraints. In this paper, we provide sufficient conditions for the existence of solutions to such systems and apply them to guarantee the non occurrence of the Lavrentieff phenomenon in optimal control under state constraints.