Conservation law constrained optimization based upon front-tracking
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 5, pp. 939-960.

We consider models based on conservation laws. For the optimization of such systems, a sensitivity analysis is essential to determine how changes in the decision variables influence the objective function. Here we study the sensitivity with respect to the initial data of objective functions that depend upon the solution of Riemann problems with piecewise linear flux functions. We present representations for the one-sided directional derivatives of the objective functions. The results can be used in the numerical method called Front-Tracking.

DOI: 10.1051/m2an:2006037
Classification: 35Lxx,  76N15
Keywords: sensitivity calculus, front-tracking, conservation laws
@article{M2AN_2006__40_5_939_0,
     author = {Gugat, Martin and Herty, Micha\"el and Klar, Axel and Leugering, Gunter},
     title = {Conservation law constrained optimization based upon front-tracking},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     pages = {939--960},
     publisher = {EDP-Sciences},
     volume = {40},
     number = {5},
     year = {2006},
     doi = {10.1051/m2an:2006037},
     zbl = {1116.65079},
     mrnumber = {2293253},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an:2006037/}
}
TY  - JOUR
AU  - Gugat, Martin
AU  - Herty, Michaël
AU  - Klar, Axel
AU  - Leugering, Gunter
TI  - Conservation law constrained optimization based upon front-tracking
JO  - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY  - 2006
DA  - 2006///
SP  - 939
EP  - 960
VL  - 40
IS  - 5
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an:2006037/
UR  - https://zbmath.org/?q=an%3A1116.65079
UR  - https://www.ams.org/mathscinet-getitem?mr=2293253
UR  - https://doi.org/10.1051/m2an:2006037
DO  - 10.1051/m2an:2006037
LA  - en
ID  - M2AN_2006__40_5_939_0
ER  - 
%0 Journal Article
%A Gugat, Martin
%A Herty, Michaël
%A Klar, Axel
%A Leugering, Gunter
%T Conservation law constrained optimization based upon front-tracking
%J ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
%D 2006
%P 939-960
%V 40
%N 5
%I EDP-Sciences
%U https://doi.org/10.1051/m2an:2006037
%R 10.1051/m2an:2006037
%G en
%F M2AN_2006__40_5_939_0
Gugat, Martin; Herty, Michaël; Klar, Axel; Leugering, Gunter. Conservation law constrained optimization based upon front-tracking. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 5, pp. 939-960. doi : 10.1051/m2an:2006037. http://www.numdam.org/articles/10.1051/m2an:2006037/

[1] A. Aw and M. Rascle, Resurrection of second order models of traffic flow? SIAM J. Appl. Math. 60 (2000) 916-938. | Zbl

[2] A. Bressan, Hyperbolic Systems of Conservation Laws. Oxford University Press, Oxford (2000). | MR | Zbl

[3] C.M. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law. J. Math. Anal. Appl. 38 (1972) 33-41. | Zbl

[4] 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. | Zbl

[5] M. Gugat, Optimal nodal control of networked hyperbolic systems: Evaluation of derivatives. AMO Advanced Modeling and Optimization 7 (2005) 9-37.

[6] 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. | Zbl

[7] 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. | Zbl

[8] M. Gugat, M. Herty, A. Klar and G. Leugering, Optimal control for traffic flow networks. J. Optim. Theory Appl. 126 (2005) 589-616. | Zbl

[9] D. Helbing, Verkehrsdynamik. Springer-Verlag, Berlin, Heidelberg, New York (1997).

[10] R. Holdahl, H. Holden and K.-A. Lie, Unconditionally stable splitting methods for the shallow water equations. BIT 39 (1999) 451-472. | Zbl

[11] 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. | Zbl

[12] H. Holden and N.H. Risebro, A mathematical model of traffic flow on a network of unidirectional roads. SIAM J. Math. Anal. 26 (1995) 999-1017. | Zbl

[13] H. Holden and N.H. Risebro, Front tracking for hyperbolic conservation laws. Springer, New York, Berlin, Heidelberg (2002). | MR | Zbl

[14] H. Holden, L. Holden and R. Hoegh-Krohn, A numerical method for first order nonlinear scalar conservation laws in one-dimension. Comput. Math. Anal. 15 (1988) 595-602. | Zbl

[15] S.N. Kruzkov, First order quasi linear equations in several independent variables. Math. USSR Sbornik, 10 (1970) 217-243. | Zbl

[16] R.J. Leveque, Numerical methods for conservation laws. Birkhäuser Verlag, Basel, Boston, Berlin (1990). | MR | Zbl

[17] M.J. Lighthill and J.B. Whitham, On kinematic waves. Proc. Roy. Soc. Lond. A229 (1955) 281-345. | Zbl

[18] J. Smoller, Shock waves and reaction diffusion equations. Springer, New York, Berlin, Heidelberg (1994). | MR | Zbl

[19] S. Ulbrich, A sensitivity and adjoint calculus for discontinuous solutions of hyperbolic conservation laws with source terms. SIAM J. Control Optim. 41 (2002) 740-797. | Zbl

[20] S. Ulbrich, Adjoint-based derivative computations for the optimal control of discontinuous solutions of hyperbolic conservation laws. Syst. Control Lett. 3 (2003) 309-324. | Zbl

Cited by Sources: