Draft CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011 1/20 UC University of Cantabria Approximation of Elliptic Control Problems in Measure Spaces with Sparse Solutions Eduardo Casas ◭◭ University of Cantabria Santander, Spain ◮◮ eduardo.casas@unican.es ◭ ◮ A joint work with Christian Clason and Karl Kunisch (University of Graz) Back Close
Draft CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011 2/20 Sparse Controls UC min J ( u ) = 1 L 2 (Ω) + α � u � L 1 (Ω) + β 2 � y − y d � 2 2 � u � 2 L 2 (Ω) University of Cantabria ◭◭ ◮◮ ◭ ◮ Back Close
Draft CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011 2/20 Sparse Controls UC min J ( u ) = 1 L 2 (Ω) + α � u � L 1 (Ω) + β 2 � y − y d � 2 2 � u � 2 L 2 (Ω) University � − ∆ y + c 0 y = u in Ω of Cantabria y = 0 on Γ ◭◭ ◮◮ ◭ ◮ Back Close
Draft CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011 2/20 Sparse Controls UC min J ( u ) = 1 L 2 (Ω) + α � u � L 1 (Ω) + β 2 � y − y d � 2 2 � u � 2 L 2 (Ω) University � − ∆ y + c 0 y = u in Ω of Cantabria y = 0 on Γ u ( x ) = − 1 If α = 0 and β > 0 ⇒ ¯ β ¯ ϕ ( x ) , x ∈ Ω ◭◭ ◮◮ ◭ ◮ Back Close
Draft CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011 2/20 Sparse Controls UC min J ( u ) = 1 L 2 (Ω) + α � u � L 1 (Ω) + β 2 � y − y d � 2 2 � u � 2 L 2 (Ω) University � − ∆ y + c 0 y = u in Ω of Cantabria y = 0 on Γ u ( x ) = − 1 If α = 0 and β > 0 ⇒ ¯ β ¯ ϕ ( x ) , x ∈ Ω If α > 0 and β > 0 ⇒ supp(¯ u ) ⊂ { x ∈ Ω : | ¯ ϕ ( x ) | ≥ α } ◭◭ ◮◮ ◭ ◮ Back Close
Draft CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011 2/20 Sparse Controls UC min J ( u ) = 1 L 2 (Ω) + α � u � L 1 (Ω) + β 2 � y − y d � 2 2 � u � 2 L 2 (Ω) University � − ∆ y + c 0 y = u in Ω of Cantabria y = 0 on Γ u ( x ) = − 1 If α = 0 and β > 0 ⇒ ¯ β ¯ ϕ ( x ) , x ∈ Ω If α > 0 and β > 0 ⇒ supp(¯ u ) ⊂ { x ∈ Ω : | ¯ ϕ ( x ) | ≥ α } ◭◭ ◮◮ ◭ If α > 0 and β = 0 ⇒ supp(¯ u ) ⊂ { x ∈ Ω : | ¯ ϕ ( x ) | = α } . ◮ Back Close
Draft CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011 3/20 Setting of the Control Problem (P) UC University u ∈M (Ω) J ( u ) = 1 2 � y − y d � 2 of Cantabria (P) min L 2 (Ω) + α � u � M (Ω) , ◭◭ ◮◮ ◭ ◮ Back Close
Draft CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011 3/20 Setting of the Control Problem (P) UC University u ∈M (Ω) J ( u ) = 1 2 � y − y d � 2 of Cantabria (P) min L 2 (Ω) + α � u � M (Ω) , � − ∆ y + c 0 y = u in Ω , (1) y = 0 on Γ , ◭◭ ◮◮ ◭ ◮ Back Close
Draft CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011 3/20 Setting of the Control Problem (P) UC University u ∈M (Ω) J ( u ) = 1 2 � y − y d � 2 of Cantabria (P) min L 2 (Ω) + α � u � M (Ω) , � − ∆ y + c 0 y = u in Ω , (1) y = 0 on Γ , with c 0 ∈ L ∞ (Ω) and c 0 ≥ 0 . We assume that α > 0 , y d ∈ L 2 (Ω) and Ω is a bounded domain in R n , n = 2 or 3 , which is supposed to either be convex or have a C 1 , 1 boundary Γ . The controls are taken in the space of regular Borel measures M (Ω) . ◭◭ ◮◮ ◭ ◮ Back Close
Draft CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011 3/20 Setting of the Control Problem (P) UC University u ∈M (Ω) J ( u ) = 1 2 � y − y d � 2 of Cantabria (P) min L 2 (Ω) + α � u � M (Ω) , � − ∆ y + c 0 y = u in Ω , (1) y = 0 on Γ , with c 0 ∈ L ∞ (Ω) and c 0 ≥ 0 . We assume that α > 0 , y d ∈ L 2 (Ω) and Ω is a bounded domain in R n , n = 2 or 3 , which is supposed to either be convex or have a C 1 , 1 boundary Γ . The controls are taken in the space of regular Borel measures M (Ω) . ◭◭ ◮◮ � ◭ � u � M (Ω) = sup � u, z � = sup z ( x ) du = | u | (Ω) ◮ � z � C 0(Ω) ≤ 1 � z � C 0(Ω) ≤ 1 Ω Back Close
Draft CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011 4/20 Related Papers UC • C. Clason and K. Kunisch: “A duality-based approach to elliptic control University problems in non-reflexive Banach spaces”, ESAIM Control Optim. Calc. of Cantabria Var., 17:1 (2011), pp. 243–266. • E.C., R. Herzog and G. Wachsmuth: “Optimality conditions and error analysis of semilinear elliptic control problems with L 1 cost functional”. Submitted. • G. Stadler: “Elliptic optimal control problems with L 1 -control cost and applications for the placement of control devices”, Comp. Optim. Appls., 44:2 (2009), pp. 159–181. ◭◭ ◮◮ • D. Wachsmuth and G. Wachsmuth: “Convergence and regularization ◭ results for optimal control problems with sparsity functional”, ESAIM ◮ Control Optim. Calc. Var. 2010, DOI: 10.1051/cocv/2010027. Back Close
Draft CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011 5/20 The State Equation UC University of Cantabria Given a measure u ∈ M (Ω) , we say that y is a solution to the state equation if � � for all z ∈ H 2 (Ω) ∩ H 1 ( − ∆ z + c 0 z ) y dx = z du 0 (Ω) Ω Ω ◭◭ ◮◮ ◭ ◮ Back Close
Draft CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011 5/20 The State Equation UC University of Cantabria Given a measure u ∈ M (Ω) , we say that y is a solution to the state equation if � � for all z ∈ H 2 (Ω) ∩ H 1 ( − ∆ z + c 0 z ) y dx = z du 0 (Ω) Ω Ω It is well known that there exists a unique solution in this sense. More- over, y ∈ W 1 ,p n 0 (Ω) for every 1 ≤ p < n − 1 and ◭◭ � y � W 1 ,p (Ω) ≤ C p � u � M (Ω) ◮◮ 0 ◭ ◮ Back Close
Draft CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011 6/20 Optimality Conditions UC University of Cantabria THEOREM 1 The problem (P) has a unique solution ¯ u . ◭◭ ◮◮ ◭ ◮ Back Close
Draft CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011 6/20 Optimality Conditions UC University of Cantabria THEOREM 1 The problem (P) has a unique solution ¯ u . Moreover, if ϕ ∈ H 2 (Ω) ∩ H 1 y denotes the associated state, and ¯ ¯ 0 (Ω) the adjoint state � − ∆ ¯ ϕ + c 0 ¯ ϕ = ¯ y − y d in Ω ϕ = ¯ 0 on Γ ◭◭ ◮◮ ◭ ◮ Back Close
Draft CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011 6/20 Optimality Conditions UC University of Cantabria THEOREM 1 The problem (P) has a unique solution ¯ u . Moreover, if ϕ ∈ H 2 (Ω) ∩ H 1 y denotes the associated state, and ¯ ¯ 0 (Ω) the adjoint state � − ∆ ¯ ϕ + c 0 ¯ ϕ = ¯ y − y d in Ω ϕ = ¯ 0 on Γ then � α � ¯ u � M (Ω) + ϕ d ¯ ¯ u = 0 , Ω ◭◭ � � ¯ ϕ � C 0 (Ω) = α if ¯ u � = 0 , ◮◮ ◭ � ¯ ϕ � C 0 (Ω) ≤ α if ¯ u = 0 . ◮ Back Close
Draft CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011 7/20 Sparsity UC University u + − ¯ u − , If we consider the Jordan decomposition of ¯ u = ¯ of Cantabria ◭◭ ◮◮ ◭ ◮ Back Close
Draft CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011 7/20 Sparsity UC University u + − ¯ u − , then If we consider the Jordan decomposition of ¯ u = ¯ of Cantabria � supp(¯ u + ) ⊂ { x ∈ Ω : ¯ ϕ ( x ) = − α } , u − ) ⊂ { x ∈ Ω : ¯ supp(¯ ϕ ( x ) = + α } . ◭◭ ◮◮ ◭ ◮ Back Close
Draft CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011 7/20 Sparsity UC University u + − ¯ u − , then If we consider the Jordan decomposition of ¯ u = ¯ of Cantabria � supp(¯ u + ) ⊂ { x ∈ Ω : ¯ ϕ ( x ) = − α } , u − ) ⊂ { x ∈ Ω : ¯ supp(¯ ϕ ( x ) = + α } . THEOREM 2 There exists ¯ α > 0 such that ¯ u = 0 for every α > ¯ α . ◭◭ ◮◮ ◭ ◮ Back Close
Draft CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011 7/20 Sparsity UC University u + − ¯ u − , then If we consider the Jordan decomposition of ¯ u = ¯ of Cantabria � supp(¯ u + ) ⊂ { x ∈ Ω : ¯ ϕ ( x ) = − α } , u − ) ⊂ { x ∈ Ω : ¯ supp(¯ ϕ ( x ) = + α } . THEOREM 2 There exists ¯ α > 0 such that ¯ u = 0 for every α > ¯ α . Proof. 1 ◭◭ 2 � y α − y d � 2 L 2 (Ω) ≤ J α ( u α ) ◮◮ ◭ ◮ Back Close
Draft CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011 7/20 Sparsity UC University u + − ¯ u − , then If we consider the Jordan decomposition of ¯ u = ¯ of Cantabria � supp(¯ u + ) ⊂ { x ∈ Ω : ¯ ϕ ( x ) = − α } , u − ) ⊂ { x ∈ Ω : ¯ supp(¯ ϕ ( x ) = + α } . THEOREM 2 There exists ¯ α > 0 such that ¯ u = 0 for every α > ¯ α . Proof. 1 ◭◭ 2 � y α − y d � 2 L 2 (Ω) ≤ J α ( u α ) ≤ J α (0) ◮◮ ◭ ◮ Back Close
Draft CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011 7/20 Sparsity UC University u + − ¯ u − , then If we consider the Jordan decomposition of ¯ u = ¯ of Cantabria � supp(¯ u + ) ⊂ { x ∈ Ω : ¯ ϕ ( x ) = − α } , u − ) ⊂ { x ∈ Ω : ¯ supp(¯ ϕ ( x ) = + α } . THEOREM 2 There exists ¯ α > 0 such that ¯ u = 0 for every α > ¯ α . Proof. 1 L 2 (Ω) ≤ J α ( u α ) ≤ J α (0) = 1 ◭◭ 2 � y α − y d � 2 2 � y d � 2 L 2 (Ω) ◮◮ ◭ ◮ Back Close
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