Stable sequential Pontryagin maximum principle in optimal control problems with phase restrictions

In this paper, we obtain optimality conditions in an optimal control problem with pointwise phase constraints of the equality and inequality types treated as constraints in a Hilbert space. The main results of this work are the regularized, stable under errors of source data, Lagrange principle and the pointwise Pontryagin maximum principle in the iterative form, which, in turn, yield a functional way of constructing a minimizing approximate solution to the problem considered.