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Control of Coupled Partial Differential Equations pp 69–94Cite as

Birkhäuser

Optimal Boundary Control in Flood Management

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Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 155))

Abstract

In active flood hazard mitigation, lateral flow withdrawal is used to reduce the impact of flood waves in rivers. Through emergency side channels, lateral outflow is generated. The optimal outflow controls the flood in such a way that the cost of the created damage is minimized. The flow is governed by a networked system of nonlinear hyperbolic partial differential equations, coupled by algebraic node conditions. Two types of integrals appear in the objective function of the corresponding optimization problem: Boundary integrals (for example, to measure the amount of water that flows out of the system into the floodplain) and distributed integrals.

For the evaluation of the derivative of the objective function, we introduce an adjoint backwards system. For the numerical solution we consider a discretized system with a consistent discretization of the continuous adjoint system, in the sense that the discrete adjoint system yields the derivatives of the discretized objective function. Numerical examples are included.

This work was supported by DFG-research cluster: real-time optimization of complex systems; grant number Le595/13-1.

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References

  1. C. Bardos, A.Y. Leroux, and J.C. Nedelec. First order quasilinear equations with boundary conditions. Comm. in Partial Differential Equations, 4:1017–1034, 1979.

    Article  MathSciNet  Google Scholar 

  2. C. Bardos and O. Pironneau. Sensitivities for Euler flows. C. R. Acad. Sci., Paris, Ser. I, 2002.

    Google Scholar 

  3. C. Bardos and O. Pironneau. Derivatives and control in the presence of shocks. Computational Fluid Dynamics Journal, 12, 2003.

    Google Scholar 

  4. J. Borggaard, J. Burns, E. Cliff, and M. Gunzburger. Sensitivity calculations for a 2d, inviscid, supersonic forebody problem. In Identification and Control in Systems Governed by Partial Differential Equations, pages 14–25. SIAM, Philadelphia, 1993.

    Google Scholar 

  5. A. Bressan. Hyperbolic Systems of conservation laws. The one-dimensional Cauchy Problem. Oxford University Press, 2000.

    Google Scholar 

  6. G.M. Coclite and B. Piccoli. Traffic flow on a road network. S.I.S.S.A., 13:1–26, 2002.

    Google Scholar 

  7. S. Scott Collis, Kaveh Ghayour, and Matthias Heinkenschloss. Optimal transpiration boundary control for aeroaccoustics. Preprint, pages 1–37, 2002.

    Google Scholar 

  8. J.M. Coron. Local controllability of a 1-d tank containing a fluid modelled by the shallow water equations. ESAIM:COCV, 8:513–554, 2002.

    MATH  Google Scholar 

  9. J.M. Coron, B. d’Andrea Novel, and G. Bastin. A Lyapunov approach to control irrigation canals modeled by Saint-Venant equations. In ECC Karlsruhe, 1999.

    Google Scholar 

  10. C.M. Dafermos. Hyperbolic Conservation Laws in Continuum Physics. Springer, Berlin, 2000.

    Book  Google Scholar 

  11. B. de Saint-Venant. Théorie du mouvement non-permanent des eaux avec application aux crués des rivières et á l’introduction des marées dans leur lit. Comptes Rendus Academie des Sciences, 73:148–154,237–240, 1871.

    MATH  Google Scholar 

  12. M. Gugat. Boundary controllability between sub-and supercritical flow. SIAM Journal on Control and Optimization, 42:1056–1070, 2003.

    Article  MathSciNet  Google Scholar 

  13. M. Gugat. Nodal control of conservation laws on networks. In Control and Boundary Analysis. Cagnol, John (ed.) et al., Chapman & Hall/CRC, Boca Raton, FL. Lecture Notes in Pure and Applied Mathematics 240, 201–215, 2005.

    Chapter  Google Scholar 

  14. M. Gugat. Optimal nodal control of networked hyperbolic systems: Evaluation of derivatives. Advanced Modeling and Optimization, 7:9–37, 2005.

    MathSciNet  MATH  Google Scholar 

  15. M. Gugat, M. Herty, A. Klar, and G. Leugering. Optimal control for traffic flow networks. J. Optimization Theory Appl. 126:589–616, 2005.

    Article  MathSciNet  Google Scholar 

  16. M. Gugat and G. Leugering. Global boundary controllability of the de Saint-Venant equations between steady states. Annales de l’Institut Henri Poincaré, Nonlinear Analysis, 20:1–11, 2003.

    Article  Google Scholar 

  17. M. Gugat, G. Leugering, and E.J.P.G. Schmidt. Global controllability between steady supercritical flows in channel networks. Mathematical Methods in the Applied Sciences, 27:781–802, 2004.

    Article  MathSciNet  Google Scholar 

  18. Max D. Gunzburger. Perspectives in Flow Control and Optimization. SIAM, Philadelphia, 2002.

    Book  Google Scholar 

  19. B. Gustafsson, H.-O. Kreiss, and J. Oliger. Time Dependent Problems and Difference Methods. John Wiley, New York, 1995.

    MATH  Google Scholar 

  20. W.W. Hager. Rates of convergence for discrete approximations to unconstrained optimal control problems. SIAM J. Numer. Analysis, 13:449–472, 1976.

    Article  MathSciNet  Google Scholar 

  21. W.W. Hager. Runge-Kutta methods in optimal control and the transformed adjoint system. Numer. Math., 87:247–282, 2000.

    Article  MathSciNet  Google Scholar 

  22. H. Holden and N.H. Risebro. A mathematical model of traffic flow on a network of unidirectional roads. SIAM J. Math. Analysis, 26:999–1017, 1995.

    Article  MathSciNet  Google Scholar 

  23. H. Holden and N.H. Risebro. Front tracking for hyperbolic conservation laws. Springer, Berlin, 2002.

    Book  Google Scholar 

  24. C.T. Kelley. Iterative Methods for Optimization. SIAM, Philadelphia, 1999.

    Book  Google Scholar 

  25. P.G. LeFloch. Hyperbolic systems of conservation laws. The theory of classical and nonclassical shock waves. Birkhäuser, Basel, 2002.

    MATH  Google Scholar 

  26. G. Leugering and E.J.P. Georg Schmidt. On the modelling and stabilisation of flows in networks of open canals. SIAM J. on Control and Optimization, 41:164–180, 2002.

    Article  MathSciNet  Google Scholar 

  27. P.O. Lindberg and A. Wolf. Optimization of the short term operation of a cascade of hydro power stations. In William H. Hager and P.M. Pardalos, editors, Optimal control: theory, algorithms, and applications., 326–345. Kluwer Academic Publishers, Dordrecht, 1998.

    Chapter  Google Scholar 

  28. J.A. Roberson, J.J. Cassidy, and M.H. Chaudhry. Hydraulic Engineering. John Wiley, New York, 1995.

    Google Scholar 

  29. B. F. Sanders and N.D. Katapodes. Active flood hazard mitigation. ii: Bidirectional wave control. Journal of Hydraulic Engineering, 125:1057–1070, 1999.

    Article  Google Scholar 

  30. B.F. Sanders and N.D. Katapodes. Control of canal flow by adjoint sensitivity method. Journal of Irrigation and Drainage Engineering, 125:287–297, 1999.

    Article  Google Scholar 

  31. B.F. Sanders and N.D. Katapodes. Adjoint sensitivity analysis for shallow water wave control. Journal of Irrigation and Drainage Engineering, 126:909–919, 2000.

    Google Scholar 

  32. S. Shabayek, P. Steffler, and F. Hicks. Dynamic model for subcritical combining flows in channel junctions. Journal of Hydraulic Engineering, 128(9):821–828, 2002.

    Article  Google Scholar 

  33. L.G. Stanley and D.L. Steward. Design Sensitivity Analysis: Computational Issues of Sensitivity Equation Methods. SIAM, Philadelphia, 2002.

    Book  Google Scholar 

  34. J.J. Stoker. Water Waves. Interscience, New York, 1957.

    MATH  Google Scholar 

  35. Li Ta-tsien. Global classical solutions for quasilinear hyperbolic systems. Masson, Paris, 1994.

    MATH  Google Scholar 

  36. S. Ulbrich. A sensitivity and adjoint calculus for discontinuous solutions of hyperbolic conservation laws with source terms. SIAM J. Control Optim., 41:740–797, 2002.

    Article  MathSciNet  Google Scholar 

  37. S. Ulbrich. Adjoint-based derivative computations for the optimal control of discontinuous solutions of hyperbolic conservation laws. Systems and Control Letters, 48:309–324, 2003.

    MathSciNet  MATH  Google Scholar 

  38. R. Le Veque. Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge, 2002.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Institut für Angewandte Mathematik Lehrstuhl II für Angewandte Mathematik, Friedrich-Alexander-Universtität Erlangen-Nürnberg, Martensstr. 3, D-91058, Erlangen, Germany

    Martin Gugat

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  1. Martin Gugat
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Editors and Affiliations

  1. Institute for Mathematics, University of Graz, Heinrichstraße 36, A-8010, Graz, Austria

    Karl Kunisch

  2. Institute for Mathematics, Humboldt University Berlin, Unter den Linden 6, D-10099, Berlin, Germany

    Jürgen Sprekels

  3. Institute for Applied Mathematics, University of Erlangen-Nürnberg, Martensstraße 3, D-91058, Erlangen, Germany

    Günter Leugering

  4. Institute for Mathematics, Technical University Berlin, Straße des 17 Juni 136, D-10623, Berlin, Germany

    Fredi Tröltzsch

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© 2007 Birkhäuser Verlag Basel/Switzerland

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Gugat, M. (2007). Optimal Boundary Control in Flood Management. In: Kunisch, K., Sprekels, J., Leugering, G., Tröltzsch, F. (eds) Control of Coupled Partial Differential Equations. International Series of Numerical Mathematics, vol 155. Birkhäuser, Basel. https://doi.org/10.1007/978-3-7643-7721-2_4

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