Sufficient Conditions and Sensitivity Analysis for Optimal Bang-Bang Control Problems with State Constraints

Bang-bang control problems subject to a state inequality constraint are considered. It is shown that the control problem induces an optimization problem, where the optimization vector assembles the switching and junction times for bang-bang and boundary arcs. Second order sufficient conditions (SSC) for the state-constrained control problem are given which require that SSC for the induced optimization problem are satisfied and a generalized strict bang-bang property holds at switching and junction times. This type of SSC ensures solution differentiability of optimal solutions under parameter perturbations and allows to compute parametric sensitivity derivatives. A numerical algorithm is presented that simultaneously determines a solution candidate, performs the second-order test and computes parametric sensitivity derivatives. We illustrate the algorithm with two state-constrained optimal control problems in biomedicine.