Conservation law constrained optimization based upon front-tracking
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 5, p. 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 : https://doi.org/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},
publisher = {EDP-Sciences},
volume = {40},
number = {5},
year = {2006},
pages = {939-960},
doi = {10.1051/m2an:2006037},
zbl = {1116.65079},
mrnumber = {2293253},
language = {en},
url = {http://www.numdam.org/item/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/item/M2AN_2006__40_5_939_0/

[1] A. Aw and M. Rascle, Resurrection of second order models of traffic flow? SIAM J. Appl. Math. 60 (2000) 916-938. | Zbl 0957.35086

[2] A. Bressan, Hyperbolic Systems of Conservation Laws. Oxford University Press, Oxford (2000). | MR 1816648 | Zbl 0997.35002

[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 0233.35014

[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 1069.90014

[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 0987.93056

[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 1047.93028

[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 1079.49024

[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 0945.76059

[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 0851.65064

[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 0833.35089

[13] H. Holden and N.H. Risebro, Front tracking for hyperbolic conservation laws. Springer, New York, Berlin, Heidelberg (2002). | MR 1912206 | Zbl 1006.35002

[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 0658.65085

[15] S.N. Kruzkov, First order quasi linear equations in several independent variables. Math. USSR Sbornik, 10 (1970) 217-243. | Zbl 0215.16203

[16] R.J. Leveque, Numerical methods for conservation laws. Birkhäuser Verlag, Basel, Boston, Berlin (1990). | MR 1077828 | Zbl 0723.65067

[17] M.J. Lighthill and J.B. Whitham, On kinematic waves. Proc. Roy. Soc. Lond. A229 (1955) 281-345. | Zbl 0064.20905

[18] J. Smoller, Shock waves and reaction diffusion equations. Springer, New York, Berlin, Heidelberg (1994). | MR 1301779 | Zbl 0807.35002 | Zbl 0508.35002

[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 1019.49026

[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 1157.49306 | Zbl pre05055503