Optimal Control of the Unsteady Euler Equations in 1D with Application to Ignition Overpressure Attenuation in Launch Vehicles

Abstract

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 1D is derived and utilized in an adjoint-based solution procedure for the optimal control. A novel algorithm is used to solve for the optimal control solution that satis es all necessary rst order optimality conditions while locally minimizing an appropriate cost functional. The solution procedure is sufficiently exible such that it can be used to solve other distributed optimal control problems where the state dynamics are in the form of non-linear hyperbolic systems of partial differential equations and where the initial data is given and the nal data is free at a free nal time. Distributed control solutions with certain physical constraints are calculated for attenuating blast waves similar to to those generated by Ignition Over Pressure (IOP) from the Shuttle's Solid Rocket Booster during launch. The control solutions give insight to 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.