Elliptic Optimal Design Parabolic Optimal Design Conclusion Reference OPTIMAL CONTROL PROBLEMS ON THE COEFFICIENTS FOR THE PARABOLIC EQUATIONS A. Alla May 19 th , 2010 A. Alla OPTIMAL CONTROL PROBLEMS ON THE COEFFICIENTS FOR
Elliptic Optimal Design Parabolic Optimal Design Conclusion Reference Outline Elliptic Optimal Design 1 Optimal Design in Conductivity Homogenization and H-convergence Generalized Optimal Design Problem Numerical Approximation Parabolic Optimal Design 2 Optimal Design in a Parabolic framework Optimal Conditions for Parabolic Problem 3 Conclusion Reference 4 A. Alla OPTIMAL CONTROL PROBLEMS ON THE COEFFICIENTS FOR
Elliptic Optimal Design Optimal Design in Conductivity Parabolic Optimal Design Homogenization and H-convergence Conclusion Generalized Optimal Design Problem Reference Numerical Approximation Optimal Problem min J ( χ ( x )) � Ω χ ( x )= V α Objective Function � J ( χ ( x )) := [ χ ( x ) g α ( x , u χ ( x )) + ( 1 − χ ( x )) g β ( x , u χ ( x ))] dx Ω State equation x ∈ Ω ⊂ R N − div ( A χ ( x ) ∇ u χ ( x )) = f ( x ) u χ ( x ) = 0 x ∈ ∂ Ω A. Alla OPTIMAL CONTROL PROBLEMS ON THE COEFFICIENTS FOR
Elliptic Optimal Design Optimal Design in Conductivity Parabolic Optimal Design Homogenization and H-convergence Conclusion Generalized Optimal Design Problem Reference Numerical Approximation Optimal Problem min J ( χ ( x )) � Ω χ ( x )= V α Objective Function � J ( χ ( x )) := [ χ ( x ) g α ( x , u χ ( x )) + ( 1 − χ ( x )) g β ( x , u χ ( x ))] dx Ω State equation x ∈ Ω ⊂ R N − div ( A χ ( x ) ∇ u χ ( x )) = f ( x ) u χ ( x ) = 0 x ∈ ∂ Ω A. Alla OPTIMAL CONTROL PROBLEMS ON THE COEFFICIENTS FOR
Elliptic Optimal Design Optimal Design in Conductivity Parabolic Optimal Design Homogenization and H-convergence Conclusion Generalized Optimal Design Problem Reference Numerical Approximation Optimal Problem min J ( χ ( x )) � Ω χ ( x )= V α Objective Function � J ( χ ( x )) := [ χ ( x ) g α ( x , u χ ( x )) + ( 1 − χ ( x )) g β ( x , u χ ( x ))] dx Ω State equation x ∈ Ω ⊂ R N − div ( A χ ( x ) ∇ u χ ( x )) = f ( x ) u χ ( x ) = 0 x ∈ ∂ Ω A. Alla OPTIMAL CONTROL PROBLEMS ON THE COEFFICIENTS FOR
Elliptic Optimal Design Optimal Design in Conductivity Parabolic Optimal Design Homogenization and H-convergence Conclusion Generalized Optimal Design Problem Reference Numerical Approximation Assumptions Let, Ω ⊂ R N 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. A χ ( x ) = αχ ( x ) + β ( 1 − χ ( x )) , γ = α, β x → g γ ( x , λ ) measurable ∀ λ ∈ R λ → g γ ( x , λ ) continuos a.e. x ∈ Ω | g γ ( x , λ ) | ≤ k ( x ) + C λ m with k ( x ) ∈ L 1 (Ω) , 1 ≤ m ≤ 2 N N − 2 . A. Alla OPTIMAL CONTROL PROBLEMS ON THE COEFFICIENTS FOR
Elliptic Optimal Design Optimal Design in Conductivity Parabolic Optimal Design Homogenization and H-convergence Conclusion Generalized Optimal Design Problem Reference Numerical Approximation Assumptions Let, Ω ⊂ R N 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. A χ ( x ) = αχ ( x ) + β ( 1 − χ ( x )) , γ = α, β x → g γ ( x , λ ) measurable ∀ λ ∈ R λ → g γ ( x , λ ) continuos a.e. x ∈ Ω | g γ ( x , λ ) | ≤ k ( x ) + C λ m with k ( x ) ∈ L 1 (Ω) , 1 ≤ m ≤ 2 N N − 2 . A. Alla OPTIMAL CONTROL PROBLEMS ON THE COEFFICIENTS FOR
Elliptic Optimal Design Optimal Design in Conductivity Parabolic Optimal Design Homogenization and H-convergence Conclusion Generalized Optimal Design Problem Reference Numerical Approximation Assumptions Let, Ω ⊂ R N 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. A χ ( x ) = αχ ( x ) + β ( 1 − χ ( x )) , γ = α, β x → g γ ( x , λ ) measurable ∀ λ ∈ R λ → g γ ( x , λ ) continuos a.e. x ∈ Ω | g γ ( x , λ ) | ≤ k ( x ) + C λ m with k ( x ) ∈ L 1 (Ω) , 1 ≤ m ≤ 2 N N − 2 . A. Alla OPTIMAL CONTROL PROBLEMS ON THE COEFFICIENTS FOR
Elliptic Optimal Design Optimal Design in Conductivity Parabolic Optimal Design Homogenization and H-convergence Conclusion Generalized Optimal Design Problem Reference Numerical Approximation Assumptions Let, Ω ⊂ R N 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. A χ ( x ) = αχ ( x ) + β ( 1 − χ ( x )) , γ = α, β x → g γ ( x , λ ) measurable ∀ λ ∈ R λ → g γ ( x , λ ) continuos a.e. x ∈ Ω | g γ ( x , λ ) | ≤ k ( x ) + C λ m with k ( x ) ∈ L 1 (Ω) , 1 ≤ m ≤ 2 N N − 2 . A. Alla OPTIMAL CONTROL PROBLEMS ON THE COEFFICIENTS FOR
Elliptic Optimal Design Optimal Design in Conductivity Parabolic Optimal Design Homogenization and H-convergence Conclusion Generalized Optimal Design Problem Reference Numerical Approximation Limit of the calculus of variations Minimizing sequence ( χ n ) n ≥ 1 such that: n → + ∞ J ( χ n ) = inf J ( χ ) lim + For a subsequence, there exists a limit χ ∞ such that: n →∞ χ n = χ ∞ lim n →∞ J ( χ n ) ≥ J ( χ ∞ ) . lim ⇓ χ ∞ is a minimizer of J . A. Alla OPTIMAL CONTROL PROBLEMS ON THE COEFFICIENTS FOR
Elliptic Optimal Design Optimal Design in Conductivity Parabolic Optimal Design Homogenization and H-convergence Conclusion Generalized Optimal Design Problem Reference Numerical Approximation Limit of the calculus of variations Minimizing sequence ( χ n ) n ≥ 1 such that: n → + ∞ J ( χ n ) = inf J ( χ ) lim + For a subsequence, there exists a limit χ ∞ such that: n →∞ χ n = χ ∞ lim n →∞ J ( χ n ) ≥ J ( χ ∞ ) . lim ⇓ χ ∞ is a minimizer of J . A. Alla OPTIMAL CONTROL PROBLEMS ON THE COEFFICIENTS FOR
Elliptic Optimal Design Optimal Design in Conductivity Parabolic Optimal Design Homogenization and H-convergence Conclusion Generalized Optimal Design Problem Reference Numerical Approximation PROBLEM shown by F . Murat and L. Tartar The problem is to find the right convergence for the sequence ( χ n ) n ≥ 1 , requiring to be compact and J ( χ ) to be continuous in L ∞ (Ω; { 0 , 1 } ) . strong convergence: the minimizing sequence is not 1 compact. weak * convergence: J is not continuous. 2 SOLUTION=RELAXATION To find the closure space of admissible designs 1 To extend the objective function to this closure 2 Calculus of variations works.We will show homogenization is a key tool for this relaxation. A. Alla OPTIMAL CONTROL PROBLEMS ON THE COEFFICIENTS FOR
Elliptic Optimal Design Optimal Design in Conductivity Parabolic Optimal Design Homogenization and H-convergence Conclusion Generalized Optimal Design Problem Reference Numerical Approximation PROBLEM shown by F . Murat and L. Tartar The problem is to find the right convergence for the sequence ( χ n ) n ≥ 1 , requiring to be compact and J ( χ ) to be continuous in L ∞ (Ω; { 0 , 1 } ) . strong convergence: the minimizing sequence is not 1 compact. weak * convergence: J is not continuous. 2 SOLUTION=RELAXATION To find the closure space of admissible designs 1 To extend the objective function to this closure 2 Calculus of variations works.We will show homogenization is a key tool for this relaxation. A. Alla OPTIMAL CONTROL PROBLEMS ON THE COEFFICIENTS FOR
Elliptic Optimal Design Optimal Design in Conductivity Parabolic Optimal Design Homogenization and H-convergence Conclusion Generalized Optimal Design Problem Reference Numerical Approximation Definition H-convergence For any f ( x ) ∈ H − 1 (Ω) , a sequence of matrices A n ( x ) → A ∗ ( x ) in L ∞ (Ω; M ( α, β )) in the sense of the homogenization, or H-converge, if the sequence u n ( x ) of solutions of − div ( A n ( x ) ∇ u n ( x )) = f ( x ) x ∈ Ω u n ( x ) = 0 x ∈ ∂ Ω , satisfies u n ( x ) ⇀ u ( x ) in H 1 0 (Ω) A n ( x ) ∇ u n ( x ) ⇀ A ∗ ( x ) ∇ u ( x ) in L 2 (Ω) N , A. Alla OPTIMAL CONTROL PROBLEMS ON THE COEFFICIENTS FOR
Elliptic Optimal Design Optimal Design in Conductivity Parabolic Optimal Design Homogenization and H-convergence Conclusion Generalized Optimal Design Problem Reference Numerical Approximation 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 is a square real matrix, M ξ · ξ ≥ α | ξ | 2 , M ( α, β ) = M − 1 ξ · ξ ≥ β − 1 | ξ | 2 , ∀ ξ ∈ R N . A. Alla OPTIMAL CONTROL PROBLEMS ON THE COEFFICIENTS FOR
Recommend
More recommend
