Second-order necessary optimality conditions in state constrained optimal control

We propose second-order necessary optimality conditions for optimal control problems with general state and control constraints. In particular, the only qualification assumption on the constraints is positive linear independence of the gradients of active state constraints. Furthermore we impose only weak regularity on the data, i.e. the optimal control is supposed to be merely measurable and the dynamics may be discontinuous in the time variable as well. The results are obtained by a direct method which uses local perturbations of the reference process by second-order tangent directions. In addition we show that a pointwise condition, similar to a classical inward pointing condition, guarantees at the same time normality of the maximum principle and nonemptiness of the set of strict second-order tangent directions.