Second Order Optimality Conditions for Controls with Continuous and Bang-Bang Components

Second order necessary and sufficient optimality conditions for bang-bang control problems in a very general form have been obtained in Milyutin and Osmolovskii (1998). These conditions require the positive (semi)-definiteness of a certain quadratic form on the finite-dimensional critical cone. Using a suitable transformation via a linear matrix ODE, Maurer and Osmolovskii (2003, 2004) have developed numerical methods into test the positive definiteness of the quadratic form. The second order test has been successfully applied to several numerical examples, representing different types of bangbang control problems. In the present talk we formulate a generalization of these results to optimal control problems with a control variable having two components: a continuous unconstrained control appearing nonlinearly and a bang-bang control appearing linearly and belonging to a convex polyhedron. An important example of a problem of this type is the planar Earth-Mars transfer. The talk is based on a joint work with Helmut Maurer.