One-shot methods in function space for PDE-constrained optimal control problems

Abstract

We state and analyse one-shot methods in function space for the optimal control of nonlinear partial differential equations (PDEs) that can be formulated in terms of a fixed-point operator. A general convergence theorem is proved by generalizing the previously obtained results in finite dimensions. As application examples we consider two nonlinear elliptic model problems: the stationary solid fuel ignition model and the stationary viscous Burgers equation. For these problems we present a more detailed convergence analysis of the method. The resulting algorithms are computationally implemented in combination with an adaptive mesh refinement strategy, which leads to an improvement in the performance of the one-shot approach.

Keywords:

• one-shot method
• optimal control of PDEs
• Burgers equation
• solid fuel ignition model
• adaptive optimal control

Acknowledgements

The authors gratefully acknowledge the support of BMBF Grant 03MS632D/‘DGHPOPT’.