Second Order Sufficient Optimality Conditions for a Control Problem with Continuous and Bang-Bang Control Components: Riccati Approach

摘要

Second order sufficient optimality conditions for bang-bang control problems in a very general form have been obtained in [15,21,13,12,1]. These conditions require the positive defmiteness (coercivity) of an associated quadratic form on the finite-dimensional critical cone. In the present paper, we investigate similar conditions for 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. The coercivity of the quadratic form can be verified by checking solvability of an associated matrix Riccati equation. The results are applied to an economic control problem in optimal production and maintenance, where existing sufficient conditions fail to hold.