Juan Casado-Daz University of Sevilla Model problem: , , > 0, - PowerPoint PPT Presentation

Juan Casado-Díaz University of Sevilla

Model problem: 𝛽 , 𝛾 , 𝜈 > 0, Ω ⊂ ℝ 𝑂 open, bounded, 𝑔 ∈ 𝐼 − 1 Ω 𝑗 : Ω × ℝ × ℝ 𝑂 → ℝ Carathédory functions, 𝑗 = 1,2, 𝐺 𝑗 ( 𝑦 , 𝑡 , 𝜊 ) ≤ 𝐷 1 + 𝑡 2 + 𝜊 2 𝐺 CP inf 𝐺 1 ( 𝑦 , 𝑣 , ∇𝑣 ) 𝑒𝑦 + 𝐺 2 ( 𝑦 , 𝑣 , ∇𝑣 ) 𝑒𝑦 𝜕 Ω \ 𝜕 − div 𝛽𝜓 𝜕 + 𝛾𝜓 Ω \ 𝜕 ∇𝑣 = 𝑔 in Ω 𝑣 = 0 𝑝𝑜 𝜖 Ω 𝜕 ⊂ Ω measurable, 𝜕 ≤ 𝜈 F. Murat. This problem has not a solution in general. It is interesting to work with a relaxed formulation.

F. Murat, L.Tartar. If the functional to minimize is + ℎ ( 𝑦 , 𝑣 ) 𝛽𝜓 𝜕 + 𝛾𝜓 Ω \ 𝜕 ∇𝑣 2 𝑒𝑦 𝐻 1 ( 𝑦 , 𝑣 ) 𝑒𝑦 + 𝐻 2 ( 𝑦 , 𝑣 ) 𝑒𝑦 , 𝜕 Ω \ 𝜕 Ω A relaxation of (CP) is given by RCP inf 𝜄𝐻 1 𝑦 , 𝑣 + 1 − 𝜄 𝐻 2 𝑦 , 𝑣 + ℎ ( 𝑦 , 𝑣 ) 𝑁∇𝑣∇𝑣 𝑒𝑦 Ω − div 𝑁∇𝑣 = 𝑔 in Ω 𝑁 ∈ 𝒧 𝜄 , 𝜄 ≤ 𝜈 𝑣 = 0 𝑝𝑜 𝜖 Ω Ω 𝒧 𝜄 set of matrices constructed via homogenization using 𝛽 with proportion 𝜄 and 𝛾 with proportion 1 − 𝜄 .

L. Tartar. K. Lurie, A. Cherkaev characterize 𝒧 𝑞 , 0 ≤ 𝑞 ≤ 1 − 1 Define 𝜇 𝑞 = 𝑞 𝛽 + 1 − 𝑞 Λ 𝑞 = 𝑞𝛽 + (1 − 𝑞 ) 𝛾 , , 𝛾 If 𝑂 ≥ 2, 𝒧 𝑞 is the set of symmetric matrices with eigenvalues satisfying 𝜇 𝑞 ≤ 𝜇 1 ≤ ⋯ ≤ 𝜇 𝑂 ≤ Λ 𝑞 𝑂 1 𝑂 − 1 1 ≤ Λ 𝑞 − 𝛽 + 𝜇 𝑞 − 𝛽 𝜇 𝑗 − 𝛽 𝑗 =1 𝑂 1 𝑂 − 1 1 ≤ 𝛾 − Λ 𝑞 + 𝛾 − 𝜇 𝑞 𝛾 − 𝜇 𝑗 𝑗 =1 For our purpose it is enough to know 𝒧 𝑞 𝜊 , 𝜊 ∈ ℝ 𝑂 𝒧 𝑞 𝜊 = 𝐶 𝜇 𝑞 + Λ 𝑞 𝜊 , Λ 𝑞 − 𝜇 𝑞 𝜊 if 𝑂 ≥ 2 2 2 𝜇 𝑞 𝜊 if 𝑂 = 1

In general (JCD, J. Couce-Calvo, J.D. Martín-Gómez) the relaxed control problem has the form inf 𝐼 𝑦 , 𝑣 , ∇𝑣 , 𝑁∇𝑣 , 𝜄 𝑒𝑦 Ω − div 𝑁∇𝑣 = 𝑔 in Ω 𝑁 ∈ 𝒧 𝜄 , 𝜄𝑒𝑦 ≤ 𝜈 , 𝑣 = 0 𝑝𝑜 𝜖 Ω Ω or equivalently inf 𝐼 𝑦 , 𝑣 , ∇𝑣 , 𝜏 , 𝜄 𝑒𝑦 Ω 1 Ω , 𝜏 ∈ 𝐿 𝜄 ∇𝑣 , 𝜄𝑒𝑦 − div 𝜏 = 𝑔 in Ω , 𝑣 ∈ 𝐼 0 ≤ 𝜈 . Ω Related results:Allaire, Bellido, Grabovski, Gutiérrez, Maestre, Munch, Pedregal, Tartar,…

Remark: If 𝑣 𝑜 , 𝜕 𝑜 are solution of − div 𝛽𝜓 𝜕 𝑜 + 𝛾𝜓 Ω \ 𝜕 𝑜 ∇𝑣 𝑜 = 𝑔 in Ω , 𝑣 𝑜 = 0 𝑝𝑜 𝜖 Ω , then for a subsequence we have ∗ 𝜄 in 𝑀 ∞ Ω 1 Ω , 𝜄 𝑜 = 𝜓 𝜕 𝑜 ⇀ 𝑣 𝑜 ⇀ 𝑣 in 𝐼 0 𝜏 𝑜 = 𝛽𝜓 𝜕 𝑜 + 𝛾𝜓 Ω \ 𝜕 𝑜 ∇𝑣 𝑜 ⇀ 𝜏 𝑗𝑜 𝑀 2 Ω 𝑂 div 𝜏 𝑜 → div 𝜏 𝑗𝑜 𝐼 − 1 Ω . 𝜐 We write 𝑣 𝑜 , 𝜏 𝑜 , 𝜄 𝑜 → 𝑣 , 𝜏 , 𝜄 .

𝑣 , 𝜏 , 𝜄 = 𝐼 ( 𝑦 , 𝑣 , ℱ ∇𝑣 , 𝜏 , 𝜄 ) 𝑒𝑦 Ω is the lower semicontinous envelope for the 𝜐 -convergence of ℱ given by ℱ 𝑣 , 𝜏 , 𝜄 = 𝐺 1 ( 𝑦 , 𝑣 , ∇𝑣 ) 𝑒𝑦 + 𝐺 2 ( 𝑦 , 𝑣 , ∇𝑣 ) 𝑒𝑦 𝜕 Ω \ 𝜕 if 𝜄 = 𝜓 𝜕 , 𝜏 = 𝛽𝜓 𝜕 + 𝛾𝜓 Ω \ 𝜕 ∇𝑣 , ℱ 𝑣 , 𝜏 , 𝜄 = + ∞ otherwise.

If 𝑂 = 1 , 1 𝑦 , 𝑡 , 𝜃 2 𝑦 , 𝑡 , 𝜃 𝐼 𝑦 , 𝑡 , 𝜊 , 𝜃 , 𝑞 = 𝑞𝐺 𝛽 + (1 − 𝑞 ) 𝐺 𝛽 𝑗𝑔 𝜃 = 𝜇 ( 𝑞 ) 𝜊 + ∞ otherwise. If 𝑂 > 1, we have  Dom 𝐼 = 𝑦 , 𝑡 , 𝜊 , 𝜃 , 𝑞 : 𝜃 ∈ 𝒧 𝑞 𝜊 .  𝐼 𝑦 , 𝑡 , 𝜊 , 𝜃 , 𝑞 ≤ 𝐷 1 + 𝑡 2 + ξ 2 + 𝜃 2  𝐼 satisfies the following convexity property 𝐼 𝑦 , 𝑡 , 𝛿𝜊 1 + 1 − 𝛿 𝜊 2 , 𝛿𝜃 1 + 1 − 𝛿 𝜃 2 , 𝛿𝑞 1 + 1 − 𝛿 𝑞 2 ) ≤ 𝛿 𝐼 𝑦 , 𝑡 , 𝜊 1 , 𝜃 1 , 𝑞 1 + 1 − 𝛿 𝐼 𝑦 , 𝜊 2 , 𝜃 2 , 𝑞 2 if 𝛿 ∈ 0,1 , 𝜊 2 − 𝜊 1 ∙ 𝜃 2 − 𝜃 1 = 0.  1 𝑦 , 𝑡 , 𝛾𝜊 − 𝜃 𝜃 − 𝛽𝜊 𝐼 𝑦 , 𝑡 , 𝜊 , 𝜃 , 𝑞 = 𝑞𝐺 𝛾 − 𝛽 𝑞 + 1 − 𝑞 𝐺 1 𝑦 , 𝑡 , 𝛾 − 𝛽 1 − 𝑞 if 𝜃 ∈ 𝜖𝒧 ( 𝑞 ) 𝜊

 If 𝐺 𝑗 𝑦 , 𝑡 , 𝜊 , 𝑗 = 1, 2, are convex in 𝜊 , we have 1 𝑦 , 𝑡 , 𝛾𝜊 − 𝜃 𝜃 − 𝛽𝜊 𝐼 𝑦 , 𝑡 , 𝜊 , 𝜃 , 𝑞 ≥𝑞𝐺 𝛾 − 𝛽 𝑞 + 1 − 𝑞 𝐺 1 𝑦 , 𝑡 , 𝛾 − 𝛽 1 − 𝑞 . Cases where 𝐼 is known 2 𝑦 , 𝑡 , 𝜊 = 𝑠 𝑦 , 𝑡 𝜊 2 𝐺 1 𝑦 , 𝑡 , 𝜊 − 𝛽 𝛾 𝑠 𝑦 , 𝑡 𝜊 2 convex in 𝜊 . 𝑅 𝑦 , 𝑡 , 𝜊 = 𝐺 ⟹ 𝐼 𝑦 , 𝑡 , 𝜊 , 𝜃 , 𝑞 = ℎ 𝑦 , 𝑡 𝜊 ∙ 𝜃 + 𝑞𝑅 𝑦 , 𝑡 , 𝜃 − 𝛾𝜊 𝑞 𝛾 − 𝛽 𝛾 It contains some cases proved by Bellido, Pedregal, Grabovsky, , … ∀ 𝑦 , 𝑡 , 𝜊 , ∃𝜂 ∈ ℝ 𝑂 such that the applications 𝑢 → 𝐺 𝑗 𝑦 , 𝑡 , 𝜊 + 𝑢𝜂 are linear. 1 𝑦 , 𝑡 , 𝛾𝜊 − 𝜃 𝜃 − 𝛽𝜊 ⇒ 𝐼 𝑦 , 𝑡 , 𝜊 , 𝜃 , 𝑞 = 𝑞𝐺 𝛾 − 𝛽 𝑞 + 1 − 𝑞 𝐺 1 𝑦 , 𝑡 , 𝛾 − 𝛽 1 − 𝑞

Numerical Aproximation JCD, C. Couce-Calvo, M. Luna-Laynez, J.D. Martín-Gómez. A discretization using an upper approximation of H 𝑦 , 𝑡 , 𝜊 , 𝜃 , 𝑞 , with Take 𝐼 𝑦 , 𝑡 , 𝜊 , 𝜃 , 𝑞 ≥ 𝐼 𝑦 , 𝑡 , 𝜊 , 𝜃 , 𝑞 𝐼 𝑦 , 𝑡 , 𝜊 , 𝛽𝜊 , 1 = 𝐺 𝑦 , 𝑡 , 𝜊 , 𝛾𝜊 , 0 = 𝐺 1 𝑦 , 𝑡 , 𝜊 , 2 𝑦 , 𝑡 , 𝜊 . 𝐼 𝐼 𝑜 ℎ of Ω For h>0, we consider a triangulation 𝒰 ℎ = 𝑈 𝑗 , ℎ 𝑗 =1 n   h      T , T measurable , T 0 , diam T h i , h i , h i , h i , h  i 1    T T 0 , if i j , i , h j , h     1 V h H and a sequence of closed subspaces 0

Discretized problem ( 𝑦 , 𝑣 ℎ , ∇𝑣 ℎ , 𝑁 ℎ ∇𝑣 ℎ , 𝜄 ℎ ) 𝑒𝑦 min 𝐼 Ω ≤ 𝜈 𝑗 , 𝑁 ℎ ∈ 𝒧 𝜄 ℎ a.e. in Ω 0 ≤ 𝜄 ℎ ≤ 1, 𝜄 ℎ 𝑒𝑦 Ω 𝑣 ℎ ∈ 𝑊 ℎ , 𝑁 ℎ ∇𝑣 ℎ ∙ ∇𝑤 ℎ 𝑒𝑦 = 𝑔𝑤 ℎ 𝑒𝑦 , ∀𝑤 ℎ ∈ 𝑊 ℎ Ω Ω 𝑁 ℎ , 𝜄 ℎ constants in the elements 𝑈 𝑗 , ℎ

Assumptions on 𝑊 ℎ i) 1 Ω , 𝑤 − 𝑤 ℎ 𝐼 0 ℎ→ 0 min lim 1 ( Ω ) = 0, ∀𝑤 ∈ 𝐼 0 𝑤 ℎ ∈𝑊 ℎ ii) 𝑥 ℎ 𝜒 − 𝑤 ℎ 𝐼 0 ℎ→ 0 min lim 1 Ω = 0, 𝑤 ℎ ∈𝑊 ℎ 1 Ω , ∀𝜒 ∈ 𝐷 𝑑 ∞ Ω ∀𝑥 ℎ ∈ 𝑊 ℎ bounded in 𝐼 0 iii) 𝐼 𝑦 , 𝑣 ℎ , ∇𝑣 ℎ , 𝜏 ℎ , 𝜄 ℎ 𝑒𝑦 lim ≥ 𝐼 ( 𝑦 , 𝑣 , ∇𝑣 , 𝜏 , 𝜄 ) 𝑒𝑦 ℎ→ 0 Ω Ω 1 Ω , 𝑣 ℎ ∈ 𝑊 ∀𝜏 ℎ ⇀ 𝜏 in 𝑀 2 Ω 𝑂 ∀𝑣 ℎ ⇀ 𝑣 in 𝐼 0 ℎ , ∗ 𝜄 in 𝑀 ∞ Ω , ∀𝜄 ℎ ⇀ 0 ≤ 𝜄 ℎ ≤ 1 a.e. in Ω 1 𝜏 ℎ − 𝜏 ∙ ∇𝑤 ℎ 𝑒𝑦 ℎ→ 0 max lim = 0. 𝑤 ℎ 𝐼 0 𝑤 ℎ ∈𝑊 ℎ 1 Ω Ω

  .  H 0  1 Properties i), ii), iii) are satisfied for V h If V is a usual space of finite elements, it satisfies i), ii). h V satisfies iii). In the examples where we know H, every sequence h Theorem: The discrete problem has a solution    u h M , , h h 1 Ω 𝑣 ℎ ⇀ 𝑣 in 𝐼 0 𝑁 ℎ ∇𝑣 ℎ ⇀ 𝑁∇𝑣 in 𝑀 2 Ω 𝑂 Up to a subsequence ∗ 𝜄 in 𝑀 ∞ Ω 𝜄 ℎ ⇀        inf H x , u , u , M u , dx               ( u,M , ) solution of div M u f in , u 0 on           0 1 , M K ( ) a.e. , dx                 lim H x , u , u , M u , dx H x , u , u , M u , dx . h h h h h  h 0  

Example 1: 𝑦 , 𝑡 , 𝜊 , 𝐵𝜊 , 1 = 𝐺 𝑦 , 𝑡 , 𝜊 , 𝐶𝜊 , 0 = 𝐺 1 𝑦 , 𝑡 , 𝜊 , 2 𝑦 , 𝑡 , 𝜊 𝐼 𝐼 𝑦 , 𝑡 , 𝜊 , 𝜃 , 𝑞 = + ∞ otherwise. 𝐼 In this case, we are solving a discrete version of the original (unrelaxed) problem, i.e. 1 𝑦 , 𝑣 ℎ , ∇𝑣 ℎ 𝑒𝑦 2 𝑦 , 𝑣 ℎ , ∇𝑣 ℎ 𝑒𝑦 inf 𝐺 + 𝐺 𝜕 Ω \ 𝜕 𝑣 ℎ ∈ 𝑊 ℎ 𝛽𝜓 𝜕 ℎ + 𝛾𝜓 Ω \ 𝜕 ℎ ∇𝑣 ℎ ∙ ∇𝑤 ℎ 𝑒𝑦 = 𝑔𝑤 ℎ 𝑒𝑦 , ∀𝑤 ℎ ∈ 𝑊 ℎ Ω Ω 𝜕 ℎ ≤ 𝜈 . 𝜕 ℎ a union of elements of 𝒰 ℎ ,

in the Example 2: ( 𝑂 ≥ 2) Since we know the values of 𝐼 boundary of its domain, we can take 𝑦 , 𝑡 , 𝜊 , 𝜃 , 𝑞 𝐼 1 𝑦 , 𝑡 , 𝛾𝜊 − 𝜃 𝜃 − 𝛽𝜊 = 𝑞𝐺 𝑞 𝛾 − 𝛽 + (1 − 𝑞 ) 𝐺 2 𝑦 , 𝑡 , (1 − 𝑞 ) 𝛾 − 𝛽 if 𝜃 ∈ 𝜖𝒧 ( 𝑞 ) 𝜊 𝑦 , 𝑡 , 𝜊 , 𝜃 , 𝑞 = + ∞ elsewhere . 𝐼 = 𝐼 . Clearly, when we know the function 𝐼 we can just take 𝐼