Global optimality conditions for a dynamic blocking problem
ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 1, pp. 124-156.

The paper is concerned with a class of optimal blocking problems in the plane. We consider a time dependent set R(t) ⊂ ℝ2, described as the reachable set for a differential inclusion. To restrict its growth, a barrier Γ can be constructed, in real time. This is a one-dimensional rectifiable set which blocks the trajectories of the differential inclusion. In this paper we introduce a definition of “regular strategy”, based on a careful classification of blocking arcs. Moreover, we derive local and global necessary conditions for an optimal strategy, which minimizes the total value of the burned region plus the cost of constructing the barrier. We show that a Lagrange multiplier, corresponding to the constraint on the construction speed, can be interpreted as the “instantaneous value of time”. This value, which we compute by two separate formulas, remains constant when free arcs are constructed and is monotone decreasing otherwise.

DOI : 10.1051/cocv/2010053
Classification : 49J24, 49K24
Mots-clés : dynamic blocking problem, optimality conditions, differential inclusion with obstacles 
@article{COCV_2012__18_1_124_0,
     author = {Bressan, Alberto and Wang, Tao},
     title = {Global optimality conditions for a dynamic blocking problem},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {124--156},
     publisher = {EDP-Sciences},
     volume = {18},
     number = {1},
     year = {2012},
     doi = {10.1051/cocv/2010053},
     mrnumber = {2887930},
     zbl = {1258.49030},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2010053/}
}
TY  - JOUR
AU  - Bressan, Alberto
AU  - Wang, Tao
TI  - Global optimality conditions for a dynamic blocking problem
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2012
SP  - 124
EP  - 156
VL  - 18
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2010053/
DO  - 10.1051/cocv/2010053
LA  - en
ID  - COCV_2012__18_1_124_0
ER  - 
%0 Journal Article
%A Bressan, Alberto
%A Wang, Tao
%T Global optimality conditions for a dynamic blocking problem
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2012
%P 124-156
%V 18
%N 1
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv/2010053/
%R 10.1051/cocv/2010053
%G en
%F COCV_2012__18_1_124_0
Bressan, Alberto; Wang, Tao. Global optimality conditions for a dynamic blocking problem. ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 1, pp. 124-156. doi : 10.1051/cocv/2010053. http://www.numdam.org/articles/10.1051/cocv/2010053/

[1] J.P. Aubin and A. Cellina, Differential inclusions. Springer-Verlag, Berlin (1984). | MR | Zbl

[2] M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser, Boston (1997). | MR | Zbl

[3] V.G. Boltyanskii, Sufficient conditions for optimality and the justification of the dynamic programming principle. SIAM J. Control Optim. 4 (1966) 326-361. | MR | Zbl

[4] A. Bressan, Differential inclusions and the control of forest fires. J. Differ. Equ. 243 (2007) 179-207. | MR | Zbl

[5] A. Bressan and C. De Lellis, Existence of optimal strategies for a fire confinement problem. Comm. Pure Appl. Math. 62 (2009) 789-830. | MR | Zbl

[6] A. Bressan and Y. Hong, Optimal control problems on stratified domains. NHM 2 (2007) 313-331. | MR | Zbl

[7] A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, AIMS Series in Applied Mathematics 2. AIMS, Springfield Mo. (2007). | MR | Zbl

[8] A. Bressan and T. Wang, The minimum speed for a blocking problem on the half plane. J. Math. Anal. Appl. 356 (2009) 133-144. | MR | Zbl

[9] A. Bressan and T. Wang, Equivalent formulation and numerical analysis of a fire confinement problem. ESAIM : COCV 16 (2010) 974-1001. | Numdam | MR | Zbl

[10] A. Bressan, M. Burago, A. Friend and J. Jou, Blocking strategies for a fire control problem. Analysis and Applications 6 (2008) 229-246. | MR | Zbl

[11] P. Brunovský, Every normal linear system has a regular time-optimal synthesis. Math. Slovaca 28 (1978) 81-100. | MR | Zbl

[12] F.H. Clarke, Optimization and Nonsmooth Analysis. Second edition, SIAM, Philadelphia (1990). | MR | Zbl

[13] W.H. Fleming and R.W. Rishel, Deterministic and Stochastic Optimal Control. Springer-Verlag, New York (1975). | MR | Zbl

[14] A.D. Ioffe and V.M. Tihomirov, Theory of Extremal Problems. North-Holland, New York (1974) 233-235. | MR | Zbl

[15] R. Vinter, Optimal Control. Birkhäuser, Boston (2000). | MR | Zbl

Cité par Sources :