OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

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Here is another case, with a single servicepole at the origin, with two residence polesat (L, H 1 ) and (L, H 2 ), with a network Σ.H 2ΣL H 1The optimal pricing policy B(t) is then{B2 (t) in [0, T ]B(t) =B 2 (T ) − B 1 (T ) + B 1 (t) in [T, L] .1.210.80.60.40.20.5 1 1.5 2The case L = 2, H 1 = 0.5, H 2 = 2.35
Problem 3 - Optimal City StructuresWe consider the following model for the optimalplanning of an urban area (Buttazzo-Santambrogio SIAM M.A. 2005).• Ω the geographical region or urban areaa compact regular domain of R N• f + the density of residentsa probability measure on Ω• f − the density of servicesa probability measure on Ω.Here the distance d in Ω is fixed (for simplicitywe take the Euclidean one) while theunknowns are f + and f − that have to bedetermined in an optimal way taking intoaccount the following facts:36
Page 1 and 2: OPTIMIZATION PROBLEMS INMASS TRANSP
Page 3 and 4: The Monge problem is thenmin{ ∫ d
Page 5 and 6: Then the class of admissible transp
Page 7 and 8: Monge-Kantorovich problem:{ ∫ }mi
Page 9 and 10: Shape optimization problemsThere is
Page 12 and 13: The shape optimization problem abov
Page 14 and 15: To overcome some of these difficult
Page 16 and 17: In the mass optimization problem th
Page 18 and 19: More precisely, we consider the geo
Page 20 and 21: produces an optimal µ which has a
Page 22 and 23: Variational integrals w.r.t. a meas
Page 24 and 25: Here are some cases where the optim
Page 26 and 27: -2 -1.5 -1 -0.5 0 0.5 1 1.5 21.510.
Page 28 and 29: Problem 1 - Optimal NetworksWe cons
Page 30 and 31: •Large town policy: several ticke
Page 32 and 33: It is interesting to study the opti
Page 34 and 35: The function B can be seen as a con
Page 38 and 39: • there is a transportation cost
Page 40 and 41: We have then the optimization probl
Page 42 and 43: and so the value of the cost F (a).
Page 44 and 45: is a delicate matter. Indeed, maxim
Page 46 and 47: Some Numerical ComputationsHere are
Page 48 and 49: Optimal sets of length 0.5 and 1 in
Page 50 and 51: Now we give a model that takes into
Page 52 and 53: We also give a model where the oppo
Page 54 and 55: Theorem Assume:• J is lower semic
Page 56 and 57: ∫∫ ∫J(µ) = f(µ a )dm+ f ∞
Page 58 and 59: Concentration case. The following f
Page 60 and 61: Diffusion case. The following facts
Page 62 and 63: Theorem Let (X, d, d ′ ) be a met
Page 64 and 65: • if q < 1 + 1/N for every µ 0 a

OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

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