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 p₀, 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 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.