# Conservation law constrained optimization based upon Front-Tracking

Martin Gugat; Michaël Herty; Axel Klar; Gunter Leugering

ESAIM: Mathematical Modelling and Numerical Analysis (2007)

- Volume: 40, Issue: 5, page 939-960
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topGugat, Martin, et al. "Conservation law constrained optimization based upon Front-Tracking." ESAIM: Mathematical Modelling and Numerical Analysis 40.5 (2007): 939-960. <http://eudml.org/doc/194342>.

@article{Gugat2007,

abstract = {
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.
},

author = {Gugat, Martin, Herty, Michaël, Klar, Axel, Leugering, Gunter},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Sensitivity calculus; front-tracking; conservation laws.; conservation law; sensitivity analysis; Riemann problem; Front-Tracking; optimization},

language = {eng},

month = {1},

number = {5},

pages = {939-960},

publisher = {EDP Sciences},

title = {Conservation law constrained optimization based upon Front-Tracking},

url = {http://eudml.org/doc/194342},

volume = {40},

year = {2007},

}

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

DA - 2007/1//

PB - EDP Sciences

VL - 40

IS - 5

SP - 939

EP - 960

AB -
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.

LA - eng

KW - Sensitivity calculus; front-tracking; conservation laws.; conservation law; sensitivity analysis; Riemann problem; Front-Tracking; optimization

UR - http://eudml.org/doc/194342

ER -

## References

top- A. Aw and M. Rascle, Resurrection of second order models of traffic flow? SIAM J. Appl. Math.60 (2000) 916–938. Zbl0957.35086
- A. Bressan, Hyperbolic Systems of Conservation Laws. Oxford University Press, Oxford (2000). Zbl0977.35087
- C.M. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law. J. Math. Anal. Appl.38 (1972) 33–41. Zbl0233.35014
- 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. Zbl1069.90014
- M. Gugat, Optimal nodal control of networked hyperbolic systems: Evaluation of derivatives. AMO Advanced Modeling and Optimization7 (2005) 9–37. Zbl1165.49307
- 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. Zbl0987.93056
- 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. Zbl1047.93028
- M. Gugat, M. Herty, A. Klar and G. Leugering, Optimal control for traffic flow networks. J. Optim. Theory Appl.126 (2005) 589–616. Zbl1079.49024
- D. Helbing, Verkehrsdynamik. Springer-Verlag, Berlin, Heidelberg, New York (1997).
- R. Holdahl, H. Holden and K.-A. Lie, Unconditionally stable splitting methods for the shallow water equations. BIT39 (1999) 451–472. Zbl0945.76059
- 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. Zbl0851.65064
- 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. Zbl0833.35089
- H. Holden and N.H. Risebro, Front tracking for hyperbolic conservation laws. Springer, New York, Berlin, Heidelberg (2002). Zbl1006.35002
- 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. Zbl0658.65085
- 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). Zbl0723.65067
- M.J. Lighthill and J.B. Whitham, On kinematic waves. Proc. Roy. Soc. Lond.A229 (1955) 281–345. Zbl0064.20905
- J. Smoller, Shock waves and reaction diffusion equations. Springer, New York, Berlin, Heidelberg (1994). Zbl0807.35002
- 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. Zbl1019.49026
- S. Ulbrich, Adjoint-based derivative computations for the optimal control of discontinuous solutions of hyperbolic conservation laws. Syst. Control Lett.3 (2003) 309–324. Zbl1157.49306

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.