Optimality and stability result for bang--bang optimal controls with simple and double switch behaviour

The paper considers parametric optimal control problems with bang--bang control vector function. For this problem we give regularity and second--order optimality conditions at the nominal solution which are sufficient to: (i) existence and local uniqueness of extremals, (ii) local structure stability, (iii) strong local optimality, under parameter perturbations. Here ``local'' means in a $L_\infty$--neighbourhood of the nominal trajectory, regardless of the control values. Stability results were obtained by the first author using the shooting approach, while optimality results were obtained by the other authors, using the Hamiltonian approach. The paper, combining both the approaches, allows to unify the assumptions and to close some gaps between optimality and stability results.