Method for Optimally Controlling Unsteady Shock Strength in One Dimension

This paper presents a new formulation and computational solution of an optimal control problem concerning unsteady shock wave attenuation. The adjoint system of equations for the unsteady Euler system in one dimension is derived and used in an adjoint-based solution procedure for the optimal control. A novel algorithm is used to satisfy all necessary first-order optimality conditions while locally minimizing an appropriate cost functional. Distributed control solutions with certain physical constraints are calculated for attenuating blast waves similar to those generated by ignition overpressure from the shuttle’s solid rocket booster during launch. Results are presented for attenuating shocks traveling at Mach 1.5 and 3.5 down to 85%, 80%, and 75% of the uncontrolled wave’s driving pressure. The control solutions give insight into the magnitude and location of energy dissipation necessary to decrease a given blast wave’s overpressure to a set target level over a given spatial domain while using only as much control as needed. The solution procedure is sufficiently flexible such that it can be used to solve other optimal control problems constrained by partial differential equations that admit discontinuities and have fixed initial data and free final data at a free final time.