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 ∞ to C([0, T]; L 1 loc) which ensures the existence of optimal controls under very weak assumptions.
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© 1999 Springer Basel AG
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Ulbrich, S. (1999). On the Existence and Approximation of Solutions for the Optimal Control of Nonlinear Hyperbolic Conservation Laws. In: Hoffmann, KH., Leugering, G., Tröltzsch, F., Caesar, S. (eds) Optimal Control of Partial Differential Equations. ISNM International Series of Numerical Mathematics, vol 133. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8691-8_25
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DOI: https://doi.org/10.1007/978-3-0348-8691-8_25
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9731-0
Online ISBN: 978-3-0348-8691-8
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