An Optimal Control Problem for Flows with Discontinuities

Eugene M. Cliff
Aerospace and Ocean Engineering Deptartment
Virginia Polytechnic Institute and State University

Matthias Heinkenschloss
Department of Mathematics
Virginia Polytechnic Institute and State University

Ajit R. Shenoy
Aerospace and Ocean Engineering Deptartment
Virginia Polytechnic Institute and State University

Journal of Optimization Theory and Applications, Vol 94, 1997, pp. 273-309.

Abstract

In this paper we study a design problem for a duct flow with a shock. The presence of the shock often causes numerical difficulties. Good shock capturing schemes with low continuity properties often can not be combined successfully with efficient optimization methods requiring smooth functions. A remedy studied in this paper is to introduce the shock location as an explicit variable. This allows one to fit the shock and yields a problem with sufficiently often differentiable functions. We prove the existence of optimal solutions, the \frechet differentiability, and the existence of Lagrange multipliers. In the second part we introduce and investigate the discrete problem and study the relations between the optimality conditions for the infinite dimensional problem and the discretized one. This reveals information important for the numerical solution of the problem. Numerical examples are given to demonstrate the theoretical findings.

Keywords

Optimal control, Euler flow equations, sequential quadratic programming.

AMS subject classifications

49M37, 49K15

PDF file (297kB) of the ICAM Report 95-09-02 September 1995 (revised February 1996). The JOTA paper is a shortened version of this report.