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.