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Dynamic Blocking Problems for a Model of Fire Propagation

  • Alberto BressanEmail author
Chapter
Part of the Fields Institute Communications book series (FIC, volume 66)

Abstract

This paper contains a survey of recent work on a class of dynamic blocking problems. The basic model consists of a differential inclusion describing the growth of a set in the plane. To restrain its expansion, it is assumed that barriers can be constructed, in real time. Here the issues of major interest are: (i) whether the growth of the set can be eventually blocked, and (ii) what is the optimal location of the barriers, minimizing a cost criterion. After introducing the basic definitions and concepts, the paper reviews various results on the existence or non-existence of blocking strategies. A theorem on the existence of an optimal strategy is then recalled, together with various necessary conditions for optimality. Sufficient conditions for optimality and a numerical algorithm for the computation of optimal barriers are also discussed, together with several open problems.

Notes

Acknowledgements

This work was partially supported by NSF through grant DMS 1108702 “Problems of Nonlinear Control”.

References

  1. 1.
    L. Ambrosio, N. Fusco, and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, London, 2000. zbMATHGoogle Scholar
  2. 2.
    L. Ambrosio, A. Colesanti and E. Villa, Outer Minkowski content for some classes of closed sets, Math. Ann., 342 (2008), 727–748. MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    J. P. Aubin and A. Cellina, Differential Inclusions, Springer, Berlin, 1984. zbMATHCrossRefGoogle Scholar
  4. 4.
    M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser, Boston, 1997. zbMATHCrossRefGoogle Scholar
  5. 5.
    A. Bressan, Differential inclusions and the control of forest fires, J. Differ. Equ., 243 (2007), 179–207. (Special volume in honor of A. Cellina and J. Yorke.) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    A. Bressan, M. Burago, A. Friend, and J. Jou, Blocking strategies for a fire control problem, Anal. Appl., 6 (2008), 229–246. MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    A. Bressan and C. De Lellis, Existence of optimal strategies for a fire confinement problem, Commun. Pure Appl. Math., 62 (2009), 789–830. zbMATHCrossRefGoogle Scholar
  8. 8.
    A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, American Institute of Mathematical Sciences, Springfield, 2007. zbMATHGoogle Scholar
  9. 9.
    A. Bressan and T. Wang, Equivalent formulation and numerical analysis of a fire confinement problem, ESAIM Control Optim. Calc. Var., 16 (2010), 974–1001. MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    A. Bressan and T. Wang, The minimum speed for a blocking problem on the half plane, J. Math. Anal. Appl., 356 (2009), 133–144. MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    A. Bressan and T. Wang, Global necessary conditions for a dynamic blocking problem. ESAIM Control Optim. Calc. Var., 18 (2012), 124–156. MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    A. Bressan and T. Wang, On the optimal strategy for an isotropic blocking problem. Calc. Var. PDEs, 45 (2012), 125–145. MathSciNetCrossRefGoogle Scholar
  13. 13.
    L. Cesari, Optimization Theory and Applications, Springer, Berlin, 1983. zbMATHCrossRefGoogle Scholar
  14. 14.
    A. Cianchi and N. Fusco, Functions of bounded variation and rearrangements, Arch. Rational Mech. Anal., 165 (2002), 1–40. MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    C. De Lellis and R. Robyr, Hamilton-Jacobi equations with obstacles, Arch. Rational Mech. Anal., 200 (2011), 1051–1073. zbMATHCrossRefGoogle Scholar
  16. 16.
    W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer, New York, 1975. zbMATHCrossRefGoogle Scholar
  17. 17.
    V. Mallet, D. E. Keyes, and F. E. Fendell, Modeling wildland fire propagation with level set methods, Comput. Math. Appl., 57 (2009), 1089–1101. MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    G. D. Richards, An elliptical growth model of forest fire fronts and its numerical solution, Int. J. Numer. Meth. Eng., 30 (1990), 1163–1179. zbMATHCrossRefGoogle Scholar
  19. 19.
    R. C. Rothermel, A mathematical model for predicting fire spread in wildland fuels, USDA Forest Service, Intermountain Forest and Range Experiment Station, Research Paper INT-115, Ogden, Utah, USA, 1972. Google Scholar
  20. 20.
    J. A. Sethian, Level Set Methods and Fast Marching Methods, Cambridge University Press, Cambridge, 1999. zbMATHGoogle Scholar
  21. 21.
    A. L. Sullivan, Wildland surface fire spread modelling, 1990–2007, Int. J. Wildland Fire, 18 (2009), 349–403. CrossRefGoogle Scholar
  22. 22.
    R. Vinter, Optimal Control, Birkhäuser, Boston, 2000. zbMATHGoogle Scholar
  23. 23.
    T. Wang, Optimality conditions for a blocking strategy involving delaying arcs, J. Optim. Theory Appl., 152 (2012), 307–333. MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsPenn State UniversityUniversity ParkUSA

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