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 ⊂ ℝ𝑂 open, bounded, 𝑔 ∈ 𝐼−1 Ω 𝐺

𝑗: Ω × ℝ × ℝ𝑂 → ℝ Carathédory functions, 𝑗 = 1,2,

𝐺

𝑗(𝑦, 𝑡, 𝜊) ≤ 𝐷 1 + 𝑡 2 + 𝜊 2

CP inf 𝐺

1(𝑦, 𝑣, ∇𝑣)𝑒𝑦 𝜕

+ 𝐺

2(𝑦, 𝑣, ∇𝑣)𝑒𝑦 Ω\𝜕

−div 𝛽𝜓𝜕 + 𝛾𝜓Ω\𝜕 ∇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

𝐻1(𝑦, 𝑣)𝑒𝑦

𝜕

+ 𝐻2(𝑦, 𝑣)𝑒𝑦

Ω\𝜕

+ ℎ(𝑦, 𝑣) 𝛽𝜓𝜕 + 𝛾𝜓Ω\𝜕 ∇𝑣 2𝑒𝑦

Ω

, A relaxation of (CP) is given by RCP inf 𝜄𝐻1 𝑦, 𝑣 + 1 − 𝜄 𝐻2 𝑦, 𝑣 + ℎ(𝑦, 𝑣)𝑁∇𝑣∇𝑣 𝑒𝑦

Ω

−div 𝑁∇𝑣 = 𝑔 in Ω 𝑣 = 0 𝑝𝑜 𝜖 Ω 𝑁 ∈ 𝒧 𝜄 , 𝜄

Ω

≤ 𝜈 𝒧 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

Define 𝜇 𝑞 = 𝑞 𝛽 + 1 − 𝑞 𝛾

−1

, Λ 𝑞 = 𝑞𝛽 + (1 − 𝑞)𝛾, If 𝑂 ≥ 2, 𝒧 𝑞 is the set of symmetric matrices with eigenvalues satisfying 𝜇 𝑞 ≤ 𝜇1 ≤ ⋯ ≤ 𝜇𝑂 ≤ Λ 𝑞 1 𝜇𝑗 − 𝛽

𝑂 𝑗=1

≤ 𝑂 − 1 Λ 𝑞 − 𝛽 + 1 𝜇 𝑞 − 𝛽 1 𝛾 − 𝜇𝑗

𝑂 𝑗=1

≤ 𝑂 − 1 𝛾 − Λ 𝑞 + 1 𝛾 − 𝜇 𝑞 For our purpose it is enough to know 𝒧 𝑞 𝜊, 𝜊 ∈ ℝ𝑂 𝒧 𝑞 𝜊 = 𝐶 𝜇 𝑞 + Λ 𝑞 2 𝜊, Λ 𝑞 − 𝜇 𝑞 2 𝜊 if 𝑂 ≥ 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 𝑣 = 0 𝑝𝑜 𝜖 Ω 𝑁 ∈ 𝒧 𝜄 , 𝜄𝑒𝑦

Ω

≤ 𝜈,

• r equivalently inf 𝐼 𝑦, 𝑣, ∇𝑣, 𝜏, 𝜄 𝑒𝑦

Ω

−div𝜏 = 𝑔 in Ω, 𝑣 ∈ 𝐼0

1 Ω , 𝜏 ∈ 𝐿 𝜄 ∇𝑣, 𝜄𝑒𝑦 Ω

≤ 𝜈. Related results:Allaire, Bellido, Grabovski, Gutiérrez, Maestre, Munch, Pedregal, Tartar,…

Remark: If 𝑣𝑜, 𝜕𝑜 are solution of −div 𝛽𝜓𝜕𝑜 + 𝛾𝜓Ω\𝜕𝑜 ∇𝑣𝑜 = 𝑔 in Ω, 𝑣𝑜 = 0 𝑝𝑜 𝜖 Ω, then for a subsequence we have 𝑣𝑜 ⇀ 𝑣 in 𝐼0

1 Ω , 𝜄𝑜 = 𝜓𝜕𝑜 ⇀ ∗ 𝜄 in 𝑀∞ Ω

𝜏𝑜 = 𝛽𝜓𝜕𝑜 + 𝛾𝜓Ω\𝜕𝑜 ∇𝑣𝑜 ⇀ 𝜏 𝑗𝑜 𝑀2 Ω 𝑂 div𝜏𝑜 → div𝜏 𝑗𝑜 𝐼−1 Ω . We write 𝑣𝑜, 𝜏𝑜, 𝜄𝑜

𝜐

→ 𝑣, 𝜏, 𝜄 .

ℱ 𝑣, 𝜏, 𝜄 = 𝐼(𝑦, 𝑣,

Ω

∇𝑣, 𝜏, 𝜄)𝑒𝑦 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𝜕 + 𝛾𝜓Ω\𝜕 ∇𝑣, ℱ 𝑣, 𝜏, 𝜄 = +∞ otherwise.

If 𝑂 = 1, 𝐼 𝑦, 𝑡, 𝜊, 𝜃, 𝑞 = 𝑞𝐺

1 𝑦, 𝑡, 𝜃

𝛽 + (1 − 𝑞)𝐺

2 𝑦, 𝑡, 𝜃

𝛽 𝑗𝑔 𝜃 = 𝜇(𝑞)𝜊 +∞

• therwise.

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, 𝛿𝜊1 + 1 − 𝛿 𝜊2, 𝛿𝜃1 + 1 − 𝛿 𝜃2 , 𝛿𝑞1 + 1 − 𝛿 𝑞2) ≤ 𝛿 𝐼 𝑦, 𝑡, 𝜊1, 𝜃1, 𝑞1 + 1 − 𝛿 𝐼 𝑦, 𝜊2, 𝜃2, 𝑞2 if 𝛿 ∈ 0,1 , 𝜊2 − 𝜊1 ∙ 𝜃2 − 𝜃1 = 0.  𝐼 𝑦, 𝑡, 𝜊, 𝜃, 𝑞 =𝑞𝐺

1 𝑦, 𝑡, 𝛾𝜊 − 𝜃

𝛾 − 𝛽 𝑞 + 1 − 𝑞 𝐺

1 𝑦, 𝑡,

𝜃 − 𝛽𝜊 𝛾 − 𝛽 1 − 𝑞 if 𝜃 ∈ 𝜖𝒧(𝑞)𝜊

 If 𝐺

𝑗 𝑦, 𝑡, 𝜊 , 𝑗 = 1, 2, are convex in 𝜊, we have

𝐼 𝑦, 𝑡, 𝜊, 𝜃, 𝑞 ≥𝑞𝐺

1 𝑦, 𝑡, 𝛾𝜊 − 𝜃

𝛾 − 𝛽 𝑞 + 1 − 𝑞 𝐺

1 𝑦, 𝑡,

𝜃 − 𝛽𝜊 𝛾 − 𝛽 1 − 𝑞 . Cases where 𝐼 is known 𝐺

2 𝑦, 𝑡, 𝜊 = 𝑠 𝑦, 𝑡 𝜊 2

𝑅 𝑦, 𝑡, 𝜊 = 𝐺

1 𝑦, 𝑡, 𝜊 − 𝛽

𝛾 𝑠 𝑦, 𝑡 𝜊 2 convex in 𝜊 . ⟹ 𝐼 𝑦, 𝑡, 𝜊, 𝜃, 𝑞 = ℎ 𝑦, 𝑡 𝛾 𝜊 ∙ 𝜃 + 𝑞𝑅 𝑦, 𝑡, 𝜃 − 𝛾𝜊 𝑞 𝛾 − 𝛽 It contains some cases proved by Bellido, Pedregal, Grabovsky, ,… ∀ 𝑦, 𝑡, 𝜊 , ∃𝜂 ∈ ℝ𝑂 such that the applications 𝑢 → 𝐺

𝑗 𝑦, 𝑡, 𝜊 + 𝑢𝜂 are linear.

⇒ 𝐼 𝑦, 𝑡, 𝜊, 𝜃, 𝑞 =𝑞𝐺

1 𝑦, 𝑡, 𝛾𝜊 − 𝜃

𝛾 − 𝛽 𝑞 + 1 − 𝑞 𝐺

1 𝑦, 𝑡,

𝜃 − 𝛽𝜊 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 Take 𝐼 𝑦, 𝑡, 𝜊, 𝜃, 𝑞 , with 𝐼 𝑦, 𝑡, 𝜊, 𝜃, 𝑞 ≥ 𝐼 𝑦, 𝑡, 𝜊, 𝜃, 𝑞 𝐼 𝑦, 𝑡, 𝜊, 𝛽𝜊, 1 = 𝐺

1 𝑦, 𝑡, 𝜊 ,

𝐼 𝑦, 𝑡, 𝜊, 𝛾𝜊, 0 = 𝐺

2 𝑦, 𝑡, 𝜊 .

For h>0, we consider a triangulation 𝒰

ℎ = 𝑈𝑗,ℎ 𝑗=1 𝑜ℎ of Ω

 

, if , diam , , measurable ,

, , , , , 1 ,

j i T T h T T T T

h j h i h i h i h i h n i h i

     







and a sequence of closed subspaces

 

 

1

H Vh

Discretized problem min 𝐼 (𝑦, 𝑣ℎ, ∇𝑣ℎ, 𝑁ℎ∇𝑣ℎ, 𝜄ℎ)𝑒𝑦

Ω

0 ≤ 𝜄ℎ ≤ 1, 𝜄ℎ𝑒𝑦

Ω

≤ 𝜈𝑗, 𝑁ℎ ∈ 𝒧 𝜄ℎ a.e. in Ω 𝑣ℎ ∈ 𝑊

ℎ, 𝑁ℎ∇𝑣ℎ ∙ ∇𝑤ℎ𝑒𝑦 = Ω

𝑔𝑤ℎ 𝑒𝑦,

Ω

∀𝑤ℎ ∈ 𝑊

ℎ

𝑁ℎ, 𝜄ℎ constants in the elements 𝑈

𝑗,ℎ

Assumptions on 𝑊

ℎ

i) lim

ℎ→0 min 𝑤ℎ∈𝑊ℎ

𝑤 − 𝑤ℎ 𝐼0

1(Ω) = 0,

∀𝑤 ∈ 𝐼0

1 Ω ,

ii) lim

ℎ→0 min 𝑤ℎ∈𝑊ℎ

𝑥ℎ𝜒 − 𝑤ℎ 𝐼0

1 Ω = 0,

∀𝑥ℎ ∈ 𝑊

ℎ bounded in 𝐼0 1 Ω , ∀𝜒 ∈ 𝐷𝑑 ∞ Ω

iii) lim

ℎ→0 𝐼 𝑦, 𝑣ℎ, ∇𝑣ℎ, 𝜏ℎ, 𝜄ℎ 𝑒𝑦 Ω

≥ 𝐼(𝑦, 𝑣, ∇𝑣, 𝜏, 𝜄)𝑒𝑦

Ω

∀𝑣ℎ ⇀ 𝑣 in 𝐼0

1 Ω , 𝑣ℎ ∈ 𝑊 ℎ,

∀𝜏ℎ ⇀ 𝜏 in 𝑀2 Ω 𝑂 ∀𝜄ℎ ⇀

∗ 𝜄 in 𝑀∞ Ω ,

0 ≤ 𝜄ℎ ≤ 1 a.e. in Ω lim

ℎ→0 max 𝑤ℎ∈𝑊ℎ

1 𝑤ℎ 𝐼0

1 Ω

𝜏ℎ − 𝜏 ∙ ∇𝑤ℎ𝑒𝑦

Ω

= 0.

Properties i), ii), iii) are satisfied for

 .

1 0 

 H Vh

If

h

V is a usual space of finite elements, it satisfies i), ii).

In the examples where we know H, every sequence

h

V satisfies iii).

Theorem: The discrete problem has a solution 



h h h M

u  , ,

Up to a subsequence 𝑣ℎ ⇀ 𝑣 in 𝐼0

1 Ω

𝑁ℎ∇𝑣ℎ ⇀ 𝑁∇𝑣 in 𝑀2 Ω 𝑂 𝜄ℎ ⇀

∗ 𝜄 in 𝑀∞ Ω

       

. , , , , , , , , lim , a.e. ) ( , 1

• n

, in div , , , , inf

• f

solution ) , (

   

    

                         dx u M u u x H dx u M u u x H dx K M u f u M dx u M u u x H u,M

h h h h h h

       

Example 1: 𝐼 𝑦, 𝑡, 𝜊, 𝐵𝜊, 1 = 𝐺

1 𝑦, 𝑡, 𝜊 ,

𝐼 𝑦, 𝑡, 𝜊, 𝐶𝜊, 0 = 𝐺

2 𝑦, 𝑡, 𝜊

𝐼 𝑦, 𝑡, 𝜊, 𝜃, 𝑞 = +∞ otherwise. In this case, we are solving a discrete version of the original (unrelaxed) problem, i.e. inf 𝐺

1 𝑦, 𝑣ℎ, ∇𝑣ℎ 𝑒𝑦 𝜕

+ 𝐺

2 𝑦, 𝑣ℎ, ∇𝑣ℎ 𝑒𝑦 Ω\𝜕

𝑣ℎ ∈ 𝑊

ℎ

𝛽𝜓𝜕ℎ + 𝛾𝜓Ω\𝜕ℎ ∇𝑣ℎ

Ω

∙ ∇𝑤ℎ𝑒𝑦 = 𝑔𝑤ℎ

Ω

𝑒𝑦, ∀𝑤ℎ ∈ 𝑊

ℎ

𝜕ℎ a union of elements of 𝒰

ℎ,

𝜕ℎ ≤ 𝜈.

Example 2: (𝑂 ≥ 2) Since we know the values of 𝐼 in the boundary of its domain, we can take 𝐼 𝑦, 𝑡, 𝜊, 𝜃, 𝑞 = 𝑞𝐺

1 𝑦, 𝑡, 𝛾𝜊 − 𝜃

𝑞 𝛾 − 𝛽 + (1 − 𝑞)𝐺

2 𝑦, 𝑡,

𝜃 − 𝛽𝜊 (1 − 𝑞) 𝛾 − 𝛽 if 𝜃 ∈ 𝜖𝒧(𝑞)𝜊 𝐼 𝑦, 𝑡, 𝜊, 𝜃, 𝑞 = +∞ elsewhere. Clearly, when we know the function 𝐼 we can just take 𝐼 = 𝐼.

A lower approximation of 𝐼

We consider 𝐼 𝑦, 𝑡, 𝜊, 𝜃, 𝑞 with 𝐼 𝑦, 𝑡, 𝜊, 𝜃, 𝑞 ≤ 𝐼 𝑦, 𝑡, 𝜊, 𝜃, 𝑞 For h>0, consider 𝒰

ℎ = 𝑈𝑗,ℎ 𝑗=1 𝑜ℎ as above and closed subspaces

𝑊

ℎ ⊂ 𝐼0 1 Ω satisfying properties i) and ii) as above and

lim

ℎ→0 𝐼 𝑦, 𝑣ℎ, ∇𝑣ℎ, 𝜏ℎ, 𝜄ℎ 𝑒𝑦 Ω

≥ 𝐼(𝑦, 𝑣, ∇𝑣, 𝜏, 𝜄)𝑒𝑦

Ω

∀𝑣ℎ ⇀ 𝑣 in 𝐼0

1 Ω , 𝑣ℎ ∈ 𝑊 ℎ,

∀𝜏ℎ ⇀ 𝜏 in 𝑀2 Ω 𝑂 ∀𝜄ℎ ⇀

∗ 𝜄 in 𝑀∞ Ω ,

0 ≤ 𝜄ℎ ≤ 1 a.e. in Ω lim

ℎ→0 max 𝑤ℎ∈𝑊ℎ

1 𝑤ℎ 𝐼0

1 Ω

𝜏ℎ − 𝜏 ∙ ∇𝑤ℎ𝑒𝑦

Ω

= 0.

Discretized problem ℭ = co 𝑁, 𝑞 : 𝑁 ∈ 𝒧(𝑞) min 𝐼(𝑦, 𝑣ℎ,

Ω

∇𝑣ℎ, 𝑁ℎ∇𝑣ℎ, 𝜄ℎ)𝑒𝑦 0 ≤ 𝜄ℎ ≤ 1, 𝜄ℎ𝑒𝑦

Ω

≤ 𝜈𝑗, 𝜄ℎ, 𝑁ℎ ∈ ℭ a.e. in Ω 𝑣ℎ ∈ 𝑊ℎ, 𝑁ℎ∇𝑣ℎ ∙ ∇𝑤ℎ𝑒𝑦 = 𝑔𝑤ℎ 𝑒𝑦, ∀

Ω Ω

𝑤ℎ ∈ 𝑊ℎ 𝑁ℎ, 𝜄ℎ constants in the elements 𝑈

𝑗,ℎ

Theorem: The discrete problem has a solution 



h h h M

u  , ,

Up to a subsequence 𝑣ℎ ⇀ 𝑣 𝑗𝑜 𝐼0

1 Ω

𝑁ℎ∇𝑣ℎ ⇀ 𝑁∇𝑣 𝑗𝑜 𝑀2 Ω 𝑂 𝜄ℎ ⇀

∗ 𝜄 𝑗𝑜 𝑀∞ Ω

       

   

    

                         dx u M u u x H dx u M u u x H dx K M u f u M dx u M u u x H M u

h h h h h h

        , , , , , , , , lim , a.e. ) ( , 1

• n

, in div , , , , inf

• f

solution ) , , (

Remark: Looking for the optimality conditions, we hope that the solution 𝑣 , 𝑁 , 𝜄 satisfies 𝑁 ∇𝑣 ∈ 𝜖𝒧 𝜄 ∇𝑣 a.e. in Ω. Thus, we need to take 𝐼 satisfying 𝐼 𝑦, 𝑡, 𝜊, 𝜃, 𝑞 = 𝐼 𝑦, 𝑡, 𝜊, 𝜃, 𝑞 if 𝜃 ∈ 𝜖𝒧 𝑞 𝜊. Example: If 𝐺

1 𝑦, 𝑡, 𝜊 , 𝐺 2 𝑦, 𝑡, 𝜊 are convex in 𝜊 take

𝐼 𝑦, 𝑡, 𝜊, 𝜃, 𝑞 = 𝐼 𝑦, 𝑡, 𝜊, 𝜃, 𝑞 = 𝑞𝐺 𝑦, 𝑡, 𝛾𝜊 − 𝜃 𝑞 𝛾 − 𝛽 + (1 − 𝑞)𝐺 𝑦, 𝑡, 𝜃 − 𝛽𝜊 (1 − 𝑞) 𝛾 − 𝛽

We have shown that we can solve numerically the control problem discretizing the unrelaxed or the relaxed problem. What is better?

• J. CD, C. Castro, M. Luna-Laynez, E. Zuazua consider the case 𝑂 = 1.

Control problem (CP) 𝐺

1, 𝐺 2 ∈ 𝑋1,∞

inf 𝐺

1 𝑦, 𝑣, 𝑒𝑣

𝑒𝑦 𝑒𝑦

𝜕

+ 𝐺

2 𝑦, 𝑣, 𝑒𝑣

𝑒𝑦 𝑒𝑦

(0,1)\𝜕

− 𝑒 𝑒𝑦 𝛽𝜓𝜕 + 𝛾𝜓(0,1)\𝜕 𝑒𝑣 𝑒𝑦 = 𝑔 in (0,1) 𝑣 0 = 𝑣(1) = 0 𝜕 ≤ 𝜈 Relaxed formulation (RCP) min 𝜄𝐺

1 𝑦, 𝑣, 𝜇(𝜄)

𝛽 𝑒𝑣 𝑒𝑦 + (1 − 𝜄)𝐺

2 𝑦, 𝑣, 𝜇(𝜄)

𝛾 𝑒𝑣 𝑒𝑦 𝑒𝑦

1

− 𝑒 𝑒𝑦 𝜇(𝜄) 𝑒𝑣 𝑒𝑦 = 𝑔 in (0,1) 𝑣 0 = 𝑣(1) = 0 𝜄

1

𝑒𝑦 ≤ 𝜈.

Discretization

We take a partition 𝒬

𝑠 and refinement 𝒭ℎ of respective diameters 𝑠 and ℎ

𝑊

ℎ = 𝑣 ∈ 𝐷0 0 0,1 : 𝑣 is affine in each interval of 𝒭ℎ

We consider the discretized problems 𝐸𝐷𝑄 min

𝐺1 𝑦, 𝑣, 𝑒𝑣 𝑒𝑦 𝑒𝑦

𝜕

+ 𝐺1 𝑦, 𝑣, 𝑒𝑣 𝑒𝑦 𝑒𝑦

(0,1)\𝜕

𝑣 ∈ 𝑊ℎ

𝛽𝜓𝜕 + 𝛾𝜓(0,1)\𝜕 𝑒𝑣

𝑒𝑦

1

𝑒𝑤 𝑒𝑦 𝑒𝑦 = 𝑔 𝑤𝑒𝑦, ∀𝑤 ∈ 𝑊ℎ

1

𝜕 is a union of intervals of 𝒬

𝑠,

𝜕 ≤ 𝜈.

(𝐸𝑆𝑄) min 𝜄𝐺

1 𝑦, 𝑣, 𝜇 𝜄

𝛽 𝑒𝑣 𝑒𝑦 + 1 − 𝜄 𝐺

2 𝑦, 𝑣, 𝜇 𝜄

𝛾 𝑒𝑣 𝑒𝑦 𝑒𝑦

𝜕

𝑣 ∈ 𝑊

ℎ

𝜇 𝜄 𝑒𝑣 𝑒𝑦

1

𝑒𝑤 𝑒𝑦 𝑒𝑦 = 𝑔 𝑤𝑒𝑦, ∀𝑤 ∈ 𝑊

ℎ 1

𝜄 is constant in the intervals of 𝒬

𝑠, 𝜄𝑒𝑦 1

≤ 𝜈

• Theorem. Taking 𝑠 = ℎ and 𝑔 ∈ 𝑋1,𝑚(0,1) we have

min 𝑆𝐷𝑄 − min(𝐸𝐷𝑄) ≤ 𝑑ℎ

𝑚+1 𝑚+2.

Taking 𝑔 ∈ 𝑀∞ 0,1 , 𝑠 = ℎ and assuming that there exists an optimal control in 𝐶𝑊 0,1 , we have min 𝑆𝐷𝑄 − min(𝐸𝑆𝑄) ≤ 𝑑ℎ.