Second Order Sufficient Conditions for Optimal Control Problems with Free Final Time: The Riccati Approach

Abstract

Second order sufficient conditions (SSC) for control problems with control-state constraints and free final time are presented. Instead of deriving such SSC from first principles, we transform the control problem with free final time into an augmented control problem with fixed final time for which well-known SSC exist. SSC are then expressed as a condition on the positive definiteness of the second variation. A convenient numerical tool for verifying this condition is based on the Riccati approach, where one has to find a bounded solution of an associated Riccati equation satisfying specific boundary conditions. The augmented Riccati equations for the augmented control problem are derived, and their modifications on the boundary of the control-state constraint are discussed. Two numerical examples, (1) the classical Earth-Mars orbit transfer in minimal time and (2) the Rayleigh problem in electrical engineering, demonstrate that the Riccati equation approach provides a viable numerical test of SSC.