Optimization methods for the verification of second order sufficient conditions for bang-bang controls

It has been common practice to find controls satisfying only necessary conditions for optimality, and then to use these controls assuming that they are (locally) optimal. However, sufficient conditions need to be used to ascertain that the control rule is optimal. Second order sufficient conditions (SSC) which have recently been derived by Agrachev, Stefani, and Zezza, and by Maurer and Osmolovskii, are a special form of sufficient conditions which are particularly suited for numerical verification. In this paper we present optimization methods and describe a numerical scheme for finding optimal bang-bang controls and verifying SSC. A straightforward transformation of the bang-bang arc durations allows one to use standard optimal control software to find the optimal arc durations as well as to check SSC. The proposed computational verification technique is illustrated on three example applications.