Sensitivity analysis of optimal control problems with bang-bang controls

We study bang-bang control problems that depend on a parameter p. For a fixed nominal parameter p(0), it is assumed that the bang-bang control has finitely many switching points and satisfies second order sufficient conditions (SSC). SSC are formulated and checked in terms of an associated finite-dimensional optimization problem w.r.t. the switching points and the free final time. We show that the nominal optimal bang-bang control can be locally embedded into a parametric family of optimal bang-bang controls where the switching points are differentiable function of the parameter. A well known sensitivity formula from optimization [7] is used to compute the parametric sensitivity derivatives of the switching points which also allows to determine the sensitivity derivatives of the optimal state trajectories.