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Technical Report (Stefan Ulbrich)

On the Existence and Approximation of Solutions

for the Optimal Control of Nonlinear Hyperbolic

Conservation Laws



Stefan Ulbrich
Technische Universität München
Lehrstuhl für Angewandte Mathematik und Mathematische Statistik

In: K.-H. Hoffmann, G. Leugering and F. Tröltzsch (eds.)
Optimal Control of Partial Differential Equations
Birkäuser Verlag, Int. Ser. Numer. Math. 133, Basel, Boston, 1999.

Abstract

Optimal control problems for possibly discontinuous entropy solutions of nonlinear multidimensional conservation laws with controls in source term and initial condition are considered. The control-to-state-mapping is analyzed by using monotone difference schemes and existence results for optimal controls are proven. Moreover, a result on the convergence of optimal solutions of finite dimensional approximations to solutions of the original problem is given. In the 1-D case the theory of compensated compactness is used to prove that the control-to-state-mapping is compact from $L^\infty$ to $C([0,T];L^1_{loc})$ which ensures the existence of optimal controls under very weak assumptions.