### 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.

Original language | English (US) |
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Title of host publication | Advances in Applied Mathematics, Modeling, and Computational Science |

Editors | Roderick Melnik |

Pages | 11-40 |

Number of pages | 30 |

DOIs | |

State | Published - Mar 1 2013 |

### Publication series

Name | Fields Institute Communications |
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Volume | 66 |

ISSN (Print) | 1069-5265 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Advances in Applied Mathematics, Modeling, and Computational Science*(pp. 11-40). (Fields Institute Communications; Vol. 66). https://doi.org/10.1007/978-1-4614-5389-5_2

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*Advances in Applied Mathematics, Modeling, and Computational Science.*Fields Institute Communications, vol. 66, pp. 11-40. https://doi.org/10.1007/978-1-4614-5389-5_2

**Dynamic blocking problems for a model of fire propagation.** / Bressan, Alberto.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

TY - CHAP

T1 - Dynamic blocking problems for a model of fire propagation

AU - Bressan, Alberto

PY - 2013/3/1

Y1 - 2013/3/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84874318798&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84874318798&partnerID=8YFLogxK

U2 - 10.1007/978-1-4614-5389-5_2

DO - 10.1007/978-1-4614-5389-5_2

M3 - Chapter

AN - SCOPUS:84874318798

SN - 9781461453888

T3 - Fields Institute Communications

SP - 11

EP - 40

BT - Advances in Applied Mathematics, Modeling, and Computational Science

A2 - Melnik, Roderick

ER -