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On the regularity of optimal dynamic blocking strategies

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Abstract

The paper studies a dynamic blocking problem, motivated by a model of optimal fire confinement. While the fire can expand with unit speed in all directions, barriers are constructed in real time. An optimal strategy is sought, minimizing the total value of the burned region, plus a construction cost. It is well known that optimal barriers exists. In general, they are a countable union of compact, connected, rectifiable sets. The main result of the present paper shows that optimal barriers are nowhere dense. The proof relies on new estimates on the reachable sets and on optimal trajectories for the fire, solving a minimum time problem in the presence of obstacles.

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References

  1. Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press (2000)

  2. Aubin, J.P., Cellina, A.: Differential inclusions. Set-Valued Maps and Viability Theory. Springer, Berlin (1984)

    Book  Google Scholar 

  3. Bressan, A.: Differential inclusions and the control of forest fires (special volume in honor of A. Cellina and J. Yorke). J. Differ. Equ. 243, 179–207 (2007)

    Article  Google Scholar 

  4. Bressan, A.: Dynamic blocking problems for a model of fire propagation. In: Melnik, R., Kotsireas, I. (eds.) Advances in Applied Mathematics, Modeling, and Computational Science. Fields Institute Communications, pp. 11–40. Springer, New York (2013)

    Google Scholar 

  5. Bressan, A., Burago, M., Friend, A., Jou, J.: Blocking strategies for a fire control problem. Anal. Appl. 6, 229–246 (2008)

    Article  MathSciNet  Google Scholar 

  6. Bressan, A., De Lellis, C.: Existence of optimal strategies for a fire confinement problem. Commun. Pure Appl. Math. 62, 789–830 (2009)

    Article  MathSciNet  Google Scholar 

  7. Bressan, A., Piccoli, B.: Introduction to the Mathematical Theory of Control. AIMS Series in Applied Mathematics, Springfield Mo. 2007

  8. Bressan, A., Wang, T.: Equivalent formulation and numerical analysis of a fire confinement problem. ESAIM; Control Optim. Calc. Var. 16, 974–1001 (2010)

    Article  MathSciNet  Google Scholar 

  9. Bressan, A., Wang, T.: The minimum speed for a blocking problem on the half plane. J. Math. Anal. Appl. 356, 133–144 (2009)

    Article  MathSciNet  Google Scholar 

  10. Bressan, A., Wang, T.: Global necessary conditions for a dynamic blocking problem. ESAIM; Control Optim. Calc. Var. 18, 124–156 (2012)

    Article  MathSciNet  Google Scholar 

  11. Bressan, A., Wang, T.: On the optimal strategy for an isotropic blocking problem. Calc. Var. PDE 45, 125–145 (2012)

    Article  MathSciNet  Google Scholar 

  12. Cesari, L.: Optimization—Theory and Applications. Springer, New York (1983)

    Book  Google Scholar 

  13. De Lellis, C., Robyr, R.: Hamilton–Jacobi equations with obstacles. Arch. Ration. Mech. Anal. 200, 1051–1073 (2011)

    Article  MathSciNet  Google Scholar 

  14. Federer, H.: Geometric Measure Theory. Springer, Berlin (1996)

    Book  Google Scholar 

  15. Fleming, W.H., Rishel, R.W.: Deterministic and Stochastic Optimal Control. Springer, New York (1975)

    Book  Google Scholar 

  16. Kim, S., Klein, R., Kübel, D., Langetepe, E., Schwarzwald, B.: Geometric firefighting in the half-plane. Comput. Geom. 95, 101728 (2021)

    Article  MathSciNet  Google Scholar 

  17. Kolmogorov, A.N., Fomin, S.V.: Introductory Real Analysis. Dover (1975)

  18. Sullivan, A.L.: Wildland surface fire spread modeling, 1990–2007. Internat. J. Wildland Fire 18, 349–403 (2009)

    Article  Google Scholar 

  19. Wang, T.: Optimality conditions for a blocking strategy involving delaying arcs. J. Optim. Theory Appl. 152, 307–333 (2012)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Penn State University, University Park, PA, 16802, USA

    Alberto Bressan & Maria Teresa Chiri

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  1. Alberto Bressan
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  2. Maria Teresa Chiri
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Correspondence to Maria Teresa Chiri.

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Communicated by A. Mondino.

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Bressan, A., Chiri, M.T. On the regularity of optimal dynamic blocking strategies. Calc. Var. 61, 36 (2022). https://doi.org/10.1007/s00526-021-02148-6

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  • DOI: https://doi.org/10.1007/s00526-021-02148-6

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