Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Erschienen in:
Introduction Kapitel lesen Erstes Kapitel lesen

2014 | OriginalPaper | Buchkapitel

Optimal Control of Nonlinear Hyperbolic Conservation Laws with Switching

verfasst von : Sebastian Pfaff, Stefan Ulbrich, Günter Leugering

Erschienen in: Trends in PDE Constrained Optimization

Verlag: Springer International Publishing

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

We consider optimal control problems governed by nonlinear hyperbolic conservation laws at junctions and analyze in particular the Fréchet-differentiability of the reduced objective functional. This is done by showing that the control-to-state mapping of the considered problems satisfies a generalized notion of differentiability. We consider both, the case where the controls are the initial and the boundary data as well as the case where the system is controlled by the switching times of the node condition. We present differentiability results for the considered problems in a quite general setting including an adjoint-based gradient representation of the reduced objective function.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren
Vorheriges Kapitel On a Fully Adaptive SQP Method for PDAE-Constrained Optimal Control Problems with Control and State Constraints
Nächstes Kapitel Elliptic Mathematical Programs with Equilibrium Constraints in Function Space: Optimality Conditions and Numerical Realization
Literatur
1.
F. Ancona, G.M. Coclite, On the attainable set for Temple class systems with boundary controls. SIAM J. Control Optim. 43(6), 2166–2190 (2005). (electronic)
2.
F. Ancona, A. Marson, On the attainable set for scalar nonlinear conservation laws with boundary control. SIAM J. Control Optim. 36(1), 290–312 (1998). (electronic)
3.
M.K. Banda, M. Herty, A. Klar, Coupling conditions for gas networks governed by the isothermal Euler equations. Netw. Heterog. Media 1(2), 295–314 (2006)CrossRefMATHMathSciNet
4.
C. Bardos, A.Y. le Roux, J.-C. Nédélec, First order quasilinear equations with boundary conditions. Commun. Partial Differ. Equ. 4(9), 1017–1034 (1979)CrossRefMATH
5.
S. Bianchini, On the shift differentiability of the flow generated by a hyperbolic system of conservation laws. Discret. Contin. Dyn. Syst. 6(2), 329–350 (2000)CrossRefMATHMathSciNet
6.
F. Bouchut, F. James, One-dimensional transport equations with discontinuous coefficients. Nonlinear Anal. 32(7), 891–933 (1998)CrossRefMATHMathSciNet
7.
A. Bressan, Hyperbolic Systems of Conservation Laws. Volume 20 of Oxford Lecture Series in Mathematics and Its Applications (Oxford University Press, Oxford, 2000). The one-dimensional Cauchy problem.
8.
A. Bressan, G. Guerra, Shift-differentiability of the flow generated by a conservation law. Discret. Contin. Dynam. Syst. 3(1), 35–58 (1997)MATHMathSciNet
9.
A. Bressan, A. Marson, A variational calculus for discontinuous solutions of systems of conservation laws. Commun. Partial Differ. Equ. 20(9–10), 1491–1552 (1995)MATHMathSciNet
10.
G. Bretti, R. Natalini, B. Piccoli, Numerical approximations of a traffic flow model on networks. Netw. Heterog. Media 1(1), 57–84 (2006)CrossRefMATHMathSciNet
11.
C. Castro, F. Palacios, E. Zuazua, An alternating descent method for the optimal control of the inviscid Burgers equation in the presence of shocks. Math. Models Methods Appl. Sci. 18(3), 369–416 (2008)CrossRefMATHMathSciNet
12.
B. Cockburn, F. Coquel, P.G. LeFloch, Convergence of the finite volume method for multidimensional conservation laws. SIAM J. Numer. Anal. 32(3), 687–705 (1995)CrossRefMATHMathSciNet
13.
14.
G.M. Coclite, K.H. Karlsen, Y.-S. Kwon, Initial-boundary value problems for conservation laws with source terms and the Degasperis-Procesi equation. J. Funct. Anal. 257(12), 3823–3857 (2009)CrossRefMATHMathSciNet
15.
R.M. Colombo, A. Groli, On the optimization of the initial boundary value problem for a conservation law. J. Math. Anal. Appl. 291(1), 82–99 (2004)CrossRefMATHMathSciNet
16.
R.M. Colombo, G. Guerra, M. Herty, V. Schleper, Optimal control in networks of pipes and canals. SIAM J. Control Optim. 48(3), 2032–2050 (2009)CrossRefMATHMathSciNet
17.
C.M. Dafermos, Generalized characteristics and the structure of solutions of hyperbolic conservation laws. Indiana Univ. Math. J. 26(6), 1097–1119 (1977)CrossRefMATHMathSciNet
18.
F. Dubois, P. LeFloch, Boundary conditions for nonlinear hyperbolic systems of conservation laws. J. Differ. Equ. 71(1), 93–122 (1988)CrossRefMATHMathSciNet
19.
M. Giles, S. Ulbrich, Convergence of linearized and adjoint approximations for discontinuous solutions of conservation laws. Part 1: linearized approximations and linearized output functionals. SIAM J. Numer. Anal. 48(3), 882–904 (2010)
20.
M. Giles, S. Ulbrich, Convergence of linearized and adjoint approximations for discontinuous solutions of conservation laws. Part 2: adjoint approximations and extensions. SIAM J. Numer. Anal. 48(3), 905–921 (2010)
21.
S. Göttlich, M. Herty, A. Klar, Modelling and optimization of supply chains on complex networks. Commun. Math. Sci. 4(2), 315–330 (2006)CrossRefMATHMathSciNet
22.
M. Gugat, M. Herty, A. Klar, G. Leugering, Optimal control for traffic flow networks. J. Optim. Theory Appl. 126(3), 589–616 (2005)CrossRefMATHMathSciNet
23.
M. Gugat, M. Herty, V. Schleper, Flow control in gas networks: exact controllability to a given demand. Math. Methods Appl. Sci. 34(7), 745–757 (2011)CrossRefMATHMathSciNet
24.
F.M. Hante, G. Leugering, T.I. Seidman, Modeling and analysis of modal switching in networked transport systems. Appl. Math. Optim. 59(2), 275–292 (2009)CrossRefMATHMathSciNet
25.
H. Holden, N.H. Risebro, A mathematical model of traffic flow on a network of unidirectional roads. SIAM J. Math. Anal. 26(4), 999–1017 (1995)CrossRefMATHMathSciNet
26.
S.N. Kružkov, Quasilinear parabolic equations and systems with two independent variables. Trudy Sem. Petrovsk. 5, 217–272 (1979)
27.
P. LeFloch, Explicit formula for scalar nonlinear conservation laws with boundary condition. Math. Methods Appl. Sci. 10(3), 265–287 (1988)CrossRefMathSciNet
28.
A.Y. le Roux, Étude du problème mixte pour une équation quasi-linéaire du premier ordre. C. R. Acad. Sci. Paris Sér. A-B 285(5), A351–A354 (1977)MathSciNet
29.
M.J. Lighthill, G.B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads. Proc. R. Soc. Lond. Ser. A. 229, 317–345 (1955)MATHMathSciNet
30.
J. Málek, J. Nečas, M. Rokyta, M. R uužička, Weak and Measure-Valued Solutions to Evolutionary PDEs. Volume 13 of Applied Mathematics and Mathematical Computation (Chapman & Hall, London, 1996)
31.
F. Otto, Initial-boundary value problem for a scalar conservation law. C. R. Acad. Sci. Paris Sér. I Math. 322(8), 729–734 (1996)MATH
32.
V. Perrollaz, Asymptotic stabilization of entropy solutions to scalar conservation laws through a stationary feedback law. Ann. Inst. H. Poincaré Anal. Non Linéaire 30(5), 879–915 (2013)CrossRefMATHMathSciNet
33.
E. Polak, Optimization. Volume 124 of Applied Mathematical Sciences (Springer, New York, 1997). Algorithms and consistent approximations
34.
35.
S. Ulbrich, On the existence and approximation of solutions for the optimal control of nonlinear hyperbolic conservation laws, in Optimal Control of Partial Differential Equations (Chemnitz, 1998). Volume 133 of International Series of Numerical Mathematics (Birkhäuser, Basel, 1999), pp. 287–299
36.
S. Ulbrich, Optimal control of nonlinear hyperbolic conservation laws with source terms. Habilitation, Zentrum Mathematik, Technische Universität München, 2001
37.
S. Ulbrich, A sensitivity and adjoint calculus for discontinuous solutions of hyperbolic conservation laws with source terms. SIAM J. Control Optim. 41(3), 740–797 (2002). (electronic)
38.
S. Ulbrich, Adjoint-based derivative computations for the optimal control of discontinuous solutions of hyperbolic conservation laws. Syst. Control Lett. 48(3–4), 313–328 (2003). Optimization and control of distributed systems
39.
A. Vasseur, Strong traces for solutions of multidimensional scalar conservation laws. Arch. Ration. Mech. Anal. 160(3), 181–193 (2001)CrossRefMATHMathSciNet
Metadaten
Titel
Optimal Control of Nonlinear Hyperbolic Conservation Laws with Switching
verfasst von
Sebastian Pfaff
Stefan Ulbrich
Günter Leugering
Copyright-Jahr
2014
Buch
Trends in PDE Constrained Optimization

Print ISBN: 978-3-319-05082-9

Electronic ISBN: 978-3-319-05083-6

Copyright-Jahr: 2014

DOI
https://doi.org/10.1007/978-3-319-05083-6_8