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Elliptic Optimal Design Parabolic Optimal Design Conclusion Reference

OPTIMAL CONTROL PROBLEMS ON THE COEFFICIENTS FOR THE PARABOLIC EQUATIONS

• A. Alla

May 19th, 2010

• A. Alla

OPTIMAL CONTROL PROBLEMS ON THE COEFFICIENTS FOR

Elliptic Optimal Design Parabolic Optimal Design Conclusion Reference

Outline

1

Elliptic Optimal Design Optimal Design in Conductivity Homogenization and H-convergence Generalized Optimal Design Problem Numerical Approximation

2

Parabolic Optimal Design Optimal Design in a Parabolic framework Optimal Conditions for Parabolic Problem

3

Conclusion

4

Reference

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OPTIMAL CONTROL PROBLEMS ON THE COEFFICIENTS FOR

Elliptic Optimal Design Parabolic Optimal Design Conclusion Reference Optimal Design in Conductivity Homogenization and H-convergence Generalized Optimal Design Problem Numerical Approximation

Optimal Problem

min

• Ω χ(x)=Vα

J(χ(x)) Objective Function J(χ(x)) :=

• Ω

[χ(x)gα(x, uχ(x)) + (1 − χ(x))gβ(x, uχ(x))] dx State equation −div(Aχ(x)∇uχ(x)) = f(x) x ∈ Ω ⊂ RN uχ(x) = 0 x ∈ ∂Ω

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OPTIMAL CONTROL PROBLEMS ON THE COEFFICIENTS FOR

Elliptic Optimal Design Parabolic Optimal Design Conclusion Reference Optimal Design in Conductivity Homogenization and H-convergence Generalized Optimal Design Problem Numerical Approximation

Optimal Problem

min

• Ω χ(x)=Vα

J(χ(x)) Objective Function J(χ(x)) :=

• Ω

[χ(x)gα(x, uχ(x)) + (1 − χ(x))gβ(x, uχ(x))] dx State equation −div(Aχ(x)∇uχ(x)) = f(x) x ∈ Ω ⊂ RN uχ(x) = 0 x ∈ ∂Ω

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OPTIMAL CONTROL PROBLEMS ON THE COEFFICIENTS FOR

Elliptic Optimal Design Parabolic Optimal Design Conclusion Reference Optimal Design in Conductivity Homogenization and H-convergence Generalized Optimal Design Problem Numerical Approximation

Optimal Problem

min

• Ω χ(x)=Vα

J(χ(x)) Objective Function J(χ(x)) :=

• Ω

[χ(x)gα(x, uχ(x)) + (1 − χ(x))gβ(x, uχ(x))] dx State equation −div(Aχ(x)∇uχ(x)) = f(x) x ∈ Ω ⊂ RN uχ(x) = 0 x ∈ ∂Ω

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Elliptic Optimal Design Parabolic Optimal Design Conclusion Reference Optimal Design in Conductivity Homogenization and H-convergence Generalized Optimal Design Problem Numerical Approximation

Assumptions

Let, Ω ⊂ RN bounded and open. This domain is occupied by two constituent α and β such that 0 < α < β < ∞ χ(x) ∈ L∞(Ω, {0, 1}) such that χ(x) = 1 if α is present at the point x, and 0 otherwise. Aχ(x) = αχ(x) + β(1 − χ(x)), γ = α, β    x → gγ(x, λ) measurable ∀λ ∈ R λ → gγ(x, λ) continuos a.e. x ∈ Ω |gγ(x, λ)| ≤ k(x) + Cλm with k(x) ∈ L1(Ω), 1 ≤ m ≤

2N N−2.

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Elliptic Optimal Design Parabolic Optimal Design Conclusion Reference Optimal Design in Conductivity Homogenization and H-convergence Generalized Optimal Design Problem Numerical Approximation

Assumptions

Let, Ω ⊂ RN bounded and open. This domain is occupied by two constituent α and β such that 0 < α < β < ∞ χ(x) ∈ L∞(Ω, {0, 1}) such that χ(x) = 1 if α is present at the point x, and 0 otherwise. Aχ(x) = αχ(x) + β(1 − χ(x)), γ = α, β    x → gγ(x, λ) measurable ∀λ ∈ R λ → gγ(x, λ) continuos a.e. x ∈ Ω |gγ(x, λ)| ≤ k(x) + Cλm with k(x) ∈ L1(Ω), 1 ≤ m ≤

2N N−2.

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OPTIMAL CONTROL PROBLEMS ON THE COEFFICIENTS FOR

Elliptic Optimal Design Parabolic Optimal Design Conclusion Reference Optimal Design in Conductivity Homogenization and H-convergence Generalized Optimal Design Problem Numerical Approximation

Assumptions

Let, Ω ⊂ RN bounded and open. This domain is occupied by two constituent α and β such that 0 < α < β < ∞ χ(x) ∈ L∞(Ω, {0, 1}) such that χ(x) = 1 if α is present at the point x, and 0 otherwise. Aχ(x) = αχ(x) + β(1 − χ(x)), γ = α, β    x → gγ(x, λ) measurable ∀λ ∈ R λ → gγ(x, λ) continuos a.e. x ∈ Ω |gγ(x, λ)| ≤ k(x) + Cλm with k(x) ∈ L1(Ω), 1 ≤ m ≤

2N N−2.

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Elliptic Optimal Design Parabolic Optimal Design Conclusion Reference Optimal Design in Conductivity Homogenization and H-convergence Generalized Optimal Design Problem Numerical Approximation

Assumptions

Let, Ω ⊂ RN bounded and open. This domain is occupied by two constituent α and β such that 0 < α < β < ∞ χ(x) ∈ L∞(Ω, {0, 1}) such that χ(x) = 1 if α is present at the point x, and 0 otherwise. Aχ(x) = αχ(x) + β(1 − χ(x)), γ = α, β    x → gγ(x, λ) measurable ∀λ ∈ R λ → gγ(x, λ) continuos a.e. x ∈ Ω |gγ(x, λ)| ≤ k(x) + Cλm with k(x) ∈ L1(Ω), 1 ≤ m ≤

2N N−2.

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Elliptic Optimal Design Parabolic Optimal Design Conclusion Reference Optimal Design in Conductivity Homogenization and H-convergence Generalized Optimal Design Problem Numerical Approximation

Limit of the calculus of variations

Minimizing sequence (χn)n≥1 such that: lim

n→+∞ J(χn) = inf J(χ)

+ For a subsequence, there exists a limit χ∞ such that: lim

n→∞ χn = χ∞

lim

n→∞ J(χn) ≥ J(χ∞).

⇓ χ∞ is a minimizer of J.

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Elliptic Optimal Design Parabolic Optimal Design Conclusion Reference Optimal Design in Conductivity Homogenization and H-convergence Generalized Optimal Design Problem Numerical Approximation

Limit of the calculus of variations

Minimizing sequence (χn)n≥1 such that: lim

n→+∞ J(χn) = inf J(χ)

+ For a subsequence, there exists a limit χ∞ such that: lim

n→∞ χn = χ∞

lim

n→∞ J(χn) ≥ J(χ∞).

⇓ χ∞ is a minimizer of J.

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Elliptic Optimal Design Parabolic Optimal Design Conclusion Reference Optimal Design in Conductivity Homogenization and H-convergence Generalized Optimal Design Problem Numerical Approximation

PROBLEM shown by F . Murat and L. Tartar The problem is to find the right convergence for the sequence (χn)n≥1, requiring to be compact and J(χ) to be continuous in L∞(Ω; {0, 1}).

1

strong convergence: the minimizing sequence is not compact.

2

weak * convergence: J is not continuous. SOLUTION=RELAXATION

1

To find the closure space of admissible designs

2

To extend the objective function to this closure Calculus of variations works.We will show homogenization is a key tool for this relaxation.

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OPTIMAL CONTROL PROBLEMS ON THE COEFFICIENTS FOR

Elliptic Optimal Design Parabolic Optimal Design Conclusion Reference Optimal Design in Conductivity Homogenization and H-convergence Generalized Optimal Design Problem Numerical Approximation

PROBLEM shown by F . Murat and L. Tartar The problem is to find the right convergence for the sequence (χn)n≥1, requiring to be compact and J(χ) to be continuous in L∞(Ω; {0, 1}).

1

strong convergence: the minimizing sequence is not compact.

2

weak * convergence: J is not continuous. SOLUTION=RELAXATION

1

To find the closure space of admissible designs

2

To extend the objective function to this closure Calculus of variations works.We will show homogenization is a key tool for this relaxation.

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Elliptic Optimal Design Parabolic Optimal Design Conclusion Reference Optimal Design in Conductivity Homogenization and H-convergence Generalized Optimal Design Problem Numerical Approximation

Definition H-convergence

For any f(x) ∈ H−1(Ω), a sequence of matrices An(x) → A∗(x) in L∞(Ω; M(α, β)) in the sense of the homogenization, or H-converge, if the sequence un(x) of solutions of −div(An(x)∇un(x)) = f(x) x ∈ Ω un(x) = 0 x ∈ ∂Ω, satisfies un(x) ⇀ u(x) in H1

0(Ω)

An(x)∇un(x) ⇀ A∗(x)∇u(x) in L2(Ω)N,

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Definition H-convergence

where u(x) is the solution of the homogenized equation −div(A∗(x)∇u(x)) = f(x) in Ω u(x) = 0 on ∂Ω. M(α, β) =    M is a square real matrix, Mξ · ξ ≥ α|ξ|2, M−1ξ · ξ ≥ β−1|ξ|2, ∀ξ ∈ RN.   

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Elliptic Optimal Design Parabolic Optimal Design Conclusion Reference Optimal Design in Conductivity Homogenization and H-convergence Generalized Optimal Design Problem Numerical Approximation

Properties of homogenization theory

Compactness Theorem, proved by F . Murat and L.Tartar ∀An(x) ∈ L∞(Ω; M(α, β)) ⇒ ∃ a subsequence An(x), such that An(x) H-converges to A∗(x) in L∞(Ω; M(α, β)). Uniqueness Theorem, proved by F . Murat and L.Tartar Let An(x), Bn(x) ∈ L∞(Ω; M(α, β)), which H-converge to A∗(x) and B∗(x). Let ω be an open subset of Ω. If An(x) = Bn(x) in ω, then A∗(x) = B∗(x) in ω. H-convergence ensures continuity of J and compactness of the minimizing sequence.

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Elliptic Optimal Design Parabolic Optimal Design Conclusion Reference Optimal Design in Conductivity Homogenization and H-convergence Generalized Optimal Design Problem Numerical Approximation

Properties of homogenization theory

Compactness Theorem, proved by F . Murat and L.Tartar ∀An(x) ∈ L∞(Ω; M(α, β)) ⇒ ∃ a subsequence An(x), such that An(x) H-converges to A∗(x) in L∞(Ω; M(α, β)). Uniqueness Theorem, proved by F . Murat and L.Tartar Let An(x), Bn(x) ∈ L∞(Ω; M(α, β)), which H-converge to A∗(x) and B∗(x). Let ω be an open subset of Ω. If An(x) = Bn(x) in ω, then A∗(x) = B∗(x) in ω. H-convergence ensures continuity of J and compactness of the minimizing sequence.

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Elliptic Optimal Design Parabolic Optimal Design Conclusion Reference Optimal Design in Conductivity Homogenization and H-convergence Generalized Optimal Design Problem Numerical Approximation

Properties of homogenization theory

Compactness Theorem, proved by F . Murat and L.Tartar ∀An(x) ∈ L∞(Ω; M(α, β)) ⇒ ∃ a subsequence An(x), such that An(x) H-converges to A∗(x) in L∞(Ω; M(α, β)). Uniqueness Theorem, proved by F . Murat and L.Tartar Let An(x), Bn(x) ∈ L∞(Ω; M(α, β)), which H-converge to A∗(x) and B∗(x). Let ω be an open subset of Ω. If An(x) = Bn(x) in ω, then A∗(x) = B∗(x) in ω. H-convergence ensures continuity of J and compactness of the minimizing sequence.

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Elliptic Optimal Design Parabolic Optimal Design Conclusion Reference Optimal Design in Conductivity Homogenization and H-convergence Generalized Optimal Design Problem Numerical Approximation

Generalized Optimal Design Problem

Assumptions

1

Aχ(x) ∈ L∞(Ω; M(α, β)),

2

Aχ(x) is obtained from a finite set materials M1, . . . , Mm, with an arbitrary rotation R(x), i.e. Aχ(x) =

m

• i=1

χi(x)RT(x)MiR(x),

3

R(x) ∈ L∞(Ω, SO(N)),

4

m

i=1 χi = 1 in Ω,

5

• Ω χi dx ≤ Vi, i = 1, . . . , m,

6

m

i=1 Vi ≥ |Ω|,

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Optimal Design Problem min

• Ω χi dx≤Vi

J(χ) Objective function J(χ) :=

• Ω

m

• i=1

χigi(x, u)

• dx,

State equation − div(Aχ(x) ∇u(x)) = f(x) x ∈ Ω, u(x) = 0 x ∈ ∂Ω.

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OPTIMAL CONTROL PROBLEMS ON THE COEFFICIENTS FOR

Elliptic Optimal Design Parabolic Optimal Design Conclusion Reference Optimal Design in Conductivity Homogenization and H-convergence Generalized Optimal Design Problem Numerical Approximation

Optimal Design Problem min

• Ω χi dx≤Vi

J(χ) Objective function J(χ) :=

• Ω

m

• i=1

χigi(x, u)

• dx,

State equation − div(Aχ(x) ∇u(x)) = f(x) x ∈ Ω, u(x) = 0 x ∈ ∂Ω.

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OPTIMAL CONTROL PROBLEMS ON THE COEFFICIENTS FOR

Elliptic Optimal Design Parabolic Optimal Design Conclusion Reference Optimal Design in Conductivity Homogenization and H-convergence Generalized Optimal Design Problem Numerical Approximation

Optimal Design Problem min

• Ω χi dx≤Vi

J(χ) Objective function J(χ) :=

• Ω

m

• i=1

χigi(x, u)

• dx,

State equation − div(Aχ(x) ∇u(x)) = f(x) x ∈ Ω, u(x) = 0 x ∈ ∂Ω.

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Relaxed Problem

Relaxed Functional J1(θ1, . . . , θm) =

• Ω

m

• i=1

θi(x)gi(x, u)

• dx,

−div(A(x)∇u(x)) = f(x) in Ω u(x) = 0 on ∂Ω. A(x) ∈ K(θ1(x), . . . , θm(x)) a.e. x ∈ Ω.

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Relaxed Problem

Relaxed Functional J1(θ1, . . . , θm) =

• Ω

m

• i=1

θi(x)gi(x, u)

• dx,

−div(A(x)∇u(x)) = f(x) in Ω u(x) = 0 on ∂Ω. A(x) ∈ K(θ1(x), . . . , θm(x)) a.e. x ∈ Ω.

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Relaxed Problem

Relaxed Functional J1(θ1, . . . , θm) =

• Ω

m

• i=1

θi(x)gi(x, u)

• dx,

−div(A(x)∇u(x)) = f(x) in Ω u(x) = 0 on ∂Ω. A(x) ∈ K(θ1(x), . . . , θm(x)) a.e. x ∈ Ω.

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θ1(x), . . . , θm(x), satisying the constraints 0 ≤ θi(x) ≤ 1, i = 1, . . . , m,

m

• i=1

θi(x) = 1 in Ω

• Ω

θi(x) dx ≤ Vi, i = 1, . . . , m. The set K(θ1, . . . , θm) is not known in general, but one can characterize the set K(θ1, . . . , θm)E = {AE : A ∈ K(θ1, . . . , θm), E ∈ RN}.

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θ1(x), . . . , θm(x), satisying the constraints 0 ≤ θi(x) ≤ 1, i = 1, . . . , m,

m

• i=1

θi(x) = 1 in Ω

• Ω

θi(x) dx ≤ Vi, i = 1, . . . , m. The set K(θ1, . . . , θm) is not known in general, but one can characterize the set K(θ1, . . . , θm)E = {AE : A ∈ K(θ1, . . . , θm), E ∈ RN}.

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Proposition, L. Tartar Assume that N ≥ 2. For any symmetric M, let λ1(M), λN(M) denote the smallest and largest eigenvalue of M, respectively. Define λ−(θ) and λ+(θ) by 1 λ−(θ) =

m

• i=1

θi λ1(Mi) λ+(θ) =

m

• i=1

θiλN(Mi). Then K(θ)E is the closed ball with diameter [λ−(θ)E, λ+(θ)E].

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Relaxed Functional J1(θ1, . . . , θm) =

• Ω

m

• i=1

θi(x)gi(x, u)

• dx,

−div(A(x)∇u(x)) = f(x) in Ω u(x) = 0 on ∂Ω. A ∈ B(θ) = {B : λ−(θ)I ≤ B ≤ λ+(θ)I}. we can replace A ∈ B(θ) by some A∗ ∈ K(θ) such that A∗∇u = A∇u a.e. in Ω, and one has a generalized solution of the initial problem.

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Elliptic Optimal Design Parabolic Optimal Design Conclusion Reference Optimal Design in Conductivity Homogenization and H-convergence Generalized Optimal Design Problem Numerical Approximation

Relaxed Functional J1(θ1, . . . , θm) =

• Ω

m

• i=1

θi(x)gi(x, u)

• dx,

−div(A(x)∇u(x)) = f(x) in Ω u(x) = 0 on ∂Ω. A ∈ B(θ) = {B : λ−(θ)I ≤ B ≤ λ+(θ)I}. we can replace A ∈ B(θ) by some A∗ ∈ K(θ) such that A∗∇u = A∇u a.e. in Ω, and one has a generalized solution of the initial problem.

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Elliptic Optimal Design Parabolic Optimal Design Conclusion Reference Optimal Design in Conductivity Homogenization and H-convergence Generalized Optimal Design Problem Numerical Approximation

Relaxed Functional J1(θ1, . . . , θm) =

• Ω

m

• i=1

θi(x)gi(x, u)

• dx,

−div(A(x)∇u(x)) = f(x) in Ω u(x) = 0 on ∂Ω. A ∈ B(θ) = {B : λ−(θ)I ≤ B ≤ λ+(θ)I}. we can replace A ∈ B(θ) by some A∗ ∈ K(θ) such that A∗∇u = A∇u a.e. in Ω, and one has a generalized solution of the initial problem.

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Elliptic Optimal Design Parabolic Optimal Design Conclusion Reference Optimal Design in Conductivity Homogenization and H-convergence Generalized Optimal Design Problem Numerical Approximation

Numerical Approximation

Gradient method for minimizing the objective function in the case of two-materials. Remark

1

Gradient method always converges to a (local) minimum,

2

Speed of convergence depends on the line search for finding a good step tk. The convergence is proved in G. Allaire, Shape optimization by the homogenization method, Springer, 2002.

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Numerical Approximation

Gradient method for minimizing the objective function in the case of two-materials. Remark

1

Gradient method always converges to a (local) minimum,

2

Speed of convergence depends on the line search for finding a good step tk. The convergence is proved in G. Allaire, Shape optimization by the homogenization method, Springer, 2002.

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Numerical Approximation

Gradient method for minimizing the objective function in the case of two-materials. Remark

1

Gradient method always converges to a (local) minimum,

2

Speed of convergence depends on the line search for finding a good step tk. The convergence is proved in G. Allaire, Shape optimization by the homogenization method, Springer, 2002.

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Optimal Design in a Parabolic framework

Let f(x, t) ∈ L2(0, T; H−1(Ω)) and u0(x) ∈ L2(Ω), such that un(x) defined by ∂tun(x, t) − div(An(x)∇un(x, t)) = f(x, t), in Ω × (0, T) un(x, t) = 0 on ∂Ω × (0, T), un(x, 0) = u0(x) in Ω × {0}, Proposition An → A∗ in the H-convergence sense= ⇒ un(x, t) → u(x, t), solution of the parabolic problem.

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Optimal Design in a Parabolic framework

Let f(x, t) ∈ L2(0, T; H−1(Ω)) and u0(x) ∈ L2(Ω), such that un(x) defined by ∂tun(x, t) − div(An(x)∇un(x, t)) = f(x, t), in Ω × (0, T) un(x, t) = 0 on ∂Ω × (0, T), un(x, 0) = u0(x) in Ω × {0}, Proposition An → A∗ in the H-convergence sense= ⇒ un(x, t) → u(x, t), solution of the parabolic problem.

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Sketch of the proof

Integrating between [τ, τ + h] un(x, τ + h) − un(x, τ) − div

• An(x)∇

τ+h

τ

un(x, s) ds

• =

= τ+h

τ

f(x, s) ds ⇓ ∂tu(x, t) − div(A(x)∇u(x, t)) = f(x, t); in Ω × (0, T), u(x, t) = 0 on ∂Ω × (0, T), u(x, 0) = u0(x) on Ω × {0}.

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Two-phase optimization problem

min

|χω(x)|=Vα

• Ω

χω(x)gα(x, u(x, T)) + (1 − χω(x))gβ(x, u(x, T)) dx. State Equation ∂tu(x, t) − div(Aχ(x)∇u(x, t)) = f(x, t) in Ω × (0, T), u(x, t) = 0 on ∂Ω × (0, T), u(x, 0) = u0(x) in Ω × {0}. where Aχ(x) := [αχω(x) + β(1 − χω(x))], χω(x) ∈ L∞(Ω, {0, 1}), and ω ⊂ Ω ⊂ RN.

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Two-phase optimization problem

min

|χω(x)|=Vα

• Ω

χω(x)gα(x, u(x, T)) + (1 − χω(x))gβ(x, u(x, T)) dx. State Equation ∂tu(x, t) − div(Aχ(x)∇u(x, t)) = f(x, t) in Ω × (0, T), u(x, t) = 0 on ∂Ω × (0, T), u(x, 0) = u0(x) in Ω × {0}. where Aχ(x) := [αχω(x) + β(1 − χω(x))], χω(x) ∈ L∞(Ω, {0, 1}), and ω ⊂ Ω ⊂ RN.

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Elliptic Optimal Design Parabolic Optimal Design Conclusion Reference Optimal Design in a Parabolic framework Optimal Conditions for Parabolic Problem

relaxed Dirichlet problem

min

|θ(x)|=Vα

• Ω

θ(x)gα(x, u(x, T)) + (1 − θ(x))gβ(x, u(x, T)) dx. ∂tu(x, t) − div(A(x)∇u(x, t)) = f(x, t) in Ω × (0, T), u(x, t) = 0 on ∂Ω × (0, T), u(x, 0) = u0(x) in Ω × {0}. where θ(x) ∈ L∞(Ω, [0, 1]), and A(x) ∈ K(θ(x)).

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OPTIMAL CONTROL PROBLEMS ON THE COEFFICIENTS FOR

Elliptic Optimal Design Parabolic Optimal Design Conclusion Reference Optimal Design in a Parabolic framework Optimal Conditions for Parabolic Problem

relaxed Dirichlet problem

min

|θ(x)|=Vα

• Ω

θ(x)gα(x, u(x, T)) + (1 − θ(x))gβ(x, u(x, T)) dx. ∂tu(x, t) − div(A(x)∇u(x, t)) = f(x, t) in Ω × (0, T), u(x, t) = 0 on ∂Ω × (0, T), u(x, 0) = u0(x) in Ω × {0}. where θ(x) ∈ L∞(Ω, [0, 1]), and A(x) ∈ K(θ(x)).

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Elliptic Optimal Design Parabolic Optimal Design Conclusion Reference Optimal Design in a Parabolic framework Optimal Conditions for Parabolic Problem

Proposition The set K(θ(x)) is the set of all symmetric matrices with eigenvalues λ1, . . . , λN satisfying: λ−

θ ≤ λi ≤ λ+ θ

∀1 ≤ i ≤ N,

N

• i=1

1 λi − α ≤ 1 λ−

θ − α + N − 1

λ+

θ − α N

• i=1

1 β − λi ≤ 1 β − λ−

θ

+ N − 1 β − λ+

θ

where λ−

θ and λ+ θ are defined by

λ−

θ =

θ α + 1 − θ β −1 and λ+

θ = θα + (1 − θ)β.

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Elliptic Optimal Design Parabolic Optimal Design Conclusion Reference Optimal Design in a Parabolic framework Optimal Conditions for Parabolic Problem

As usual, we use the Lagrange multiplier p(x, t) ∈ L2(0, T; H1

0(Ω)) to get the adjoint equation:

Adjoint Equation −∂tp(x, t) − div(A(x)∇p(x, t)) = 0 in Ω × (0, T) p(x, t) = 0 on ∂Ω × (0, T), p(x, T) = θ(x)∂ugα(x, u(x, T))+ +(1 − θ(x))∂ugβ(x, u(x, T)) in Ω × {T}.

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Elliptic Optimal Design Parabolic Optimal Design Conclusion Reference Optimal Design in a Parabolic framework Optimal Conditions for Parabolic Problem

Admissible direction lim

ε→0+

Aε(x) − A(x) ε = Ad(x), with lim

ε→0+

θε − θ ε = θd. where uε(x) solves: ∂tuε(x, t) − div(Aε(x)∇uε(x, t)) = f(x, t) in Ω × (0, T) uε(x, t) = 0 on ∂Ω × (0, T) uε(x, 0) = u0(x) in Ω × {0}.

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Elliptic Optimal Design Parabolic Optimal Design Conclusion Reference Optimal Design in a Parabolic framework Optimal Conditions for Parabolic Problem

In order to obtain optimality conditions, we compute the derivative respect to ε of the functional, when ε goes to 0+ of: J(θε, uε) =

• Ω

θε(x)gα(x, uε(x, T))+(1−θε(x))gβ(x, uε(x, T)) dx Optimality Condition

• Ω

θd(x)[gα(x, u(x, T)) − gβ(x, u(x, T))] dx −

• Ω

Ad(x) : C(x) dx ≥ 0 C(x) := T ∇u(x, t) ⊗ ∇p(x, t) dt.

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Elliptic Optimal Design Parabolic Optimal Design Conclusion Reference Optimal Design in a Parabolic framework Optimal Conditions for Parabolic Problem

In order to obtain optimality conditions, we compute the derivative respect to ε of the functional, when ε goes to 0+ of: J(θε, uε) =

• Ω

θε(x)gα(x, uε(x, T))+(1−θε(x))gβ(x, uε(x, T)) dx Optimality Condition

• Ω

θd(x)[gα(x, u(x, T)) − gβ(x, u(x, T))] dx −

• Ω

Ad(x) : C(x) dx ≥ 0 C(x) := T ∇u(x, t) ⊗ ∇p(x, t) dt.

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OPTIMAL CONTROL PROBLEMS ON THE COEFFICIENTS FOR

Elliptic Optimal Design Parabolic Optimal Design Conclusion Reference Optimal Design in a Parabolic framework Optimal Conditions for Parabolic Problem

In order to obtain optimality conditions, we compute the derivative respect to ε of the functional, when ε goes to 0+ of: J(θε, uε) =

• Ω

θε(x)gα(x, uε(x, T))+(1−θε(x))gβ(x, uε(x, T)) dx Optimality Condition

• Ω

θd(x)[gα(x, u(x, T)) − gβ(x, u(x, T))] dx −

• Ω

Ad(x) : C(x) dx ≥ 0 C(x) := T ∇u(x, t) ⊗ ∇p(x, t) dt.

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Theorem Let a1(x) ≤ . . . ≤ aN(x), and c1(x) ≤ . . . ≤ cN(x) eigenvalues

• f A(x) and C(x) =

⇒ ∃R(x) ∈ SO(N) s.t. RT(x)C(x)R(x) = diag(c1, . . . , cN) and RT(x)A(x)R(x) = diag(a1, . . . , aN) Reference

• J. Casado Diaz, J.Couce Calvo, J. D. Martin Gomez, Optimality

conditions for nonconvex multistate control problems in the coefficients, SIAM J. Conrol Optim. Vol 43, No. 1, pp 216-239.

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Elliptic Optimal Design Parabolic Optimal Design Conclusion Reference Optimal Design in a Parabolic framework Optimal Conditions for Parabolic Problem

Theorem Let a1(x) ≤ . . . ≤ aN(x), and c1(x) ≤ . . . ≤ cN(x) eigenvalues

• f A(x) and C(x) =

⇒ ∃R(x) ∈ SO(N) s.t. RT(x)C(x)R(x) = diag(c1, . . . , cN) and RT(x)A(x)R(x) = diag(a1, . . . , aN) Reference

• J. Casado Diaz, J.Couce Calvo, J. D. Martin Gomez, Optimality

conditions for nonconvex multistate control problems in the coefficients, SIAM J. Conrol Optim. Vol 43, No. 1, pp 216-239.

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Conclusion

1

We were able to find optimality conditions for an optimal control problem on the coefficients in a parabolic equation.

2

We weren’t able to find the eigenvalue in order to solve the problem numerically.

3

It is not clear the convexity of K(θ(x)). Work in Progress Numerical approximation of this parabolic control problem.

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OPTIMAL CONTROL PROBLEMS ON THE COEFFICIENTS FOR

Elliptic Optimal Design Parabolic Optimal Design Conclusion Reference

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OPTIMAL CONTROL PROBLEMS ON THE COEFFICIENTS FOR

Elliptic Optimal Design Parabolic Optimal Design Conclusion Reference

• G. Allaire, Shape optimization by the homogenization

method, Springer, 2002.

• J. Casado Diaz, J.Couce Calvo, J. D. Martin Gomez,

Optimality conditions for nonconvex multistate control problems in the coefficients, SIAM J. Conrol Optim. Vol 43,

• No. 1, pp 216-239.

F . Murat, L. Tartar, On the Control of Coefficients in Partial Differential Equations, in Topics in the Matematical of Composite Materials, Progr. Nonlinear Differential Equations Appl. 31, A. Cherkaev and R. Kohn, eds, Birkhauser, Boston, 1997, pp 1-8.

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OPTIMAL CONTROL PROBLEMS ON THE COEFFICIENTS FOR

Elliptic Optimal Design Parabolic Optimal Design Conclusion Reference

F . Murat, L. Tartar, H-convergence, in Topics in the Matematical of Composite Materials, Progr. Nonlinear Differential Equations Appl. 31, A. Cherkaev and R. Kohn, eds, Birkhauser, Boston, 1997, pp 21-43.

• L. Tartar, Estimations of Homogenized Coefficients, in

Topics in the Matematical of Composite Materials, Progr. Nonlinear Differential Equations Appl. 31, A. Cherkaev and

• R. Kohn, eds, Birkhauser, Boston, 1997, pp 21-43.
• L. Tartar, Remarks on the Homogenization Method in

Optimal Design Problems, Math. Sciences and Applications

• vol. 9, Gakkokotosho, Tokyo, Japan, (1997) 393-412.
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