Second order sufficient conditions for time-optimal bang-bang control

We study second order sufficient optimality conditions (SSC) for optimal control problems with control appearing linearly. Specifically, time-optimal bang-bang controls will be investigated. In [N. P. Osmolovskii, Sov. Phys. Dokl., 33 (1988), pp. 883-885; Theory of Higher Order Conditions in Optimal Control, Doctor of Sci. thesis, Moscow, 1988 ( in Russian); Russian J. Math. Phys., 2 (1995), pp. 487-516; Russian J. Math. Phys., 5 (1997), pp. 373-388; Proceedings of the Conference "Calculus of Variations and Optimal Control," Chapman & Hall/CRC, Boca Raton, FL, 2000, pp. 198-216; A. A. Milyutin and N. P. Osmolovskii, Calculus of Variations and Optimal Control, Transl. Math. Monogr. 180, AMS, Providence, RI, 1998], SSC have been developed in terms of the positive definiteness of a quadratic form on a critical cone or subspace. No systematical numerical methods for verifying SSC are to be found in these papers. In the present paper, we study explicit representations of the critical subspace. This leads to an easily implementable test for SSC in the case of a bang-bang control with one or two switching points. In general, we show that the quadratic form can be simplified by a transformation that uses a solution to a linear matrix differential equation. Particular conditions even allow us to convert the quadratic form to perfect squares. Three numerical examples demonstrate the numerical viability of the proposed tests for SSC.