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Conservation law constrained optimization based upon Front-Tracking

Published online by Cambridge University Press:  16 January 2007

Martin Gugat ,
Michaël Herty ,
Axel Klar  and
Gunter Leugering
Martin Gugat
Affiliation:
Institut für Angewandte Mathematik, FAU Erlangen-Nürnberg, Martensstr. 3, 91058 Erlangen, Germany.
Michaël Herty
Affiliation:
Fachbereich Mathematik, TU Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany.
Axel Klar
Affiliation:
Fachbereich Mathematik, TU Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany.
Gunter Leugering
Affiliation:
Institut für Angewandte Mathematik, FAU Erlangen-Nürnberg, Martensstr. 3, 91058 Erlangen, Germany.
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Abstract

We consider models based on conservation laws. For the optimizationof such systems, a sensitivity analysis is essential to determinehow changes in the decision variables influence the objectivefunction. Here we study the sensitivity with respect to the initialdata of objective functions that depend upon the solution of Riemannproblems with piecewise linear flux functions. We presentrepresentations for the one–sided directional derivatives of theobjective functions. The results can be used in the numerical methodcalled Front-Tracking.

Keywords

Sensitivity calculusfront-trackingconservation laws.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 40 , Issue 5 , September 2006 , pp. 939 - 960
Copyright
© EDP Sciences, SMAI, 2007

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References

Aw, A. and Rascle, M., Resurrection of second order models of traffic flow? SIAM J. Appl. Math. 60 (2000) 916–938. CrossRef
A. Bressan, Hyperbolic Systems of Conservation Laws. Oxford University Press, Oxford (2000).
Dafermos, C.M., Polygonal approximations of solutions of the initial value problem for a conservation law. J. Math. Anal. Appl. 38 (1972) 33–41. CrossRef
K. Ehrhardt and M. Steinbach, Nonlinear optimization in gas networks, in Modeling, Simulation and Optimization of Complex Processes, H.G. Bock, E. Kostina, H.X. Phu, R. Ranacher Eds. (2005) 139–148.
Gugat, M., Optimal nodal control of networked hyperbolic systems: Evaluation of derivatives. AMO Advanced Modeling and Optimization 7 (2005) 9–37.
M. Gugat, G. Leugering, K. Schittkowski and E.J.P.G. Schmidt, Modelling, stabilization and control of flow in networks of open channels, in Online optimization of large scale systems, M. Grötschel, S.O. Krumke, J. Rambau Eds., Springer (2001) 251–270.
M. Gugat, G. Leugering and E.J.P.G. Schmidt, Global controllability between steady supercritical flows in channel networks. Math. Meth. Appl. Sci. (2003) 781–802.
Gugat, M., Herty, M., Klar, A. and Leugering, G., Optimal control for traffic flow networks. J. Optim. Theory Appl. 126 (2005) 589–616. CrossRef
D. Helbing, Verkehrsdynamik. Springer-Verlag, Berlin, Heidelberg, New York (1997).
Holdahl, R., Holden, H. and Lie, K.-A., Unconditionally stable splitting methods for the shallow water equations. BIT 39 (1999) 451–472. CrossRef
H. Holden and L. Holden, On scalar conservation laws in one-dimension, in Ideas and Methods in Mathematical Analysis, Stochastics and Applications S. Albeverio, J. Fenstad, H. Holden, T. Lindstrøm Eds. (1992) 480–509.
Holden, H. and Risebro, N.H., A mathematical model of traffic flow on a network of unidirectional roads. SIAM J. Math. Anal. 26 (1995) 999–1017. CrossRef
H. Holden and N.H. Risebro, Front tracking for hyperbolic conservation laws. Springer, New York, Berlin, Heidelberg (2002).
Holden, H., Holden, L. and Hoegh-Krohn, R., A numerical method for first order nonlinear scalar conservation laws in one-dimension. Comput. Math. Anal. 15 (1988) 595–602.
S.N. Kruzkov, First order quasi linear equations in several independent variables. Math. USSR Sbornik, 10 (1970) 217–243.
R.J. LeVeque, Numerical methods for conservation laws. Birkhäuser Verlag, Basel, Boston, Berlin (1990).
Lighthill, M.J. and Whitham, J.B., On kinematic waves. Proc. Roy. Soc. Lond. A229 (1955) 281–345. CrossRef
J. Smoller, Shock waves and reaction diffusion equations. Springer, New York, Berlin, Heidelberg (1994).
Ulbrich, S., A sensitivity and adjoint calculus for discontinuous solutions of hyperbolic conservation laws with source terms. SIAM J. Control Optim. 41 (2002) 740–797. CrossRef
Ulbrich, S., Adjoint-based derivative computations for the optimal control of discontinuous solutions of hyperbolic conservation laws. Syst. Control Lett. 3 (2003) 309–324.