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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

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◭◭ ◮◮ ◭ ◮ Back Close

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) CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close Sparse Controls min J(u) = 1 2y − yd2 L2(Ω) + αuL1(Ω) + β 2u2 L2(Ω) CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close Sparse Controls min J(u) = 1 2y − yd2 L2(Ω) + αuL1(Ω) + β 2u2 L2(Ω) −∆y + c0y = u in Ω y = 0 on Γ CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close Sparse Controls min J(u) = 1 2y − yd2 L2(Ω) + αuL1(Ω) + β 2u2 L2(Ω) −∆y + c0y = u in Ω y = 0 on Γ If α = 0 and β > 0 ⇒ ¯ u(x) = −1 β ¯ ϕ(x), x ∈ Ω CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close Sparse Controls min J(u) = 1 2y − yd2 L2(Ω) + αuL1(Ω) + β 2u2 L2(Ω) −∆y + c0y = u in Ω y = 0 on Γ If α = 0 and β > 0 ⇒ ¯ u(x) = −1 β ¯ ϕ(x), x ∈ Ω If α > 0 and β > 0 ⇒ supp(¯ u) ⊂ {x ∈ Ω : | ¯ ϕ(x)| ≥ α} CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close Sparse Controls min J(u) = 1 2y − yd2 L2(Ω) + αuL1(Ω) + β 2u2 L2(Ω) −∆y + c0y = u in Ω y = 0 on Γ If α = 0 and β > 0 ⇒ ¯ u(x) = −1 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}. CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close Setting of the Control Problem (P) (P) min u∈M(Ω) J(u) = 1 2y − yd2 L2(Ω) + αuM(Ω), CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close Setting of the Control Problem (P) (P) min u∈M(Ω) J(u) = 1 2y − yd2 L2(Ω) + αuM(Ω), −∆y + c0y = u in Ω, y = 0 on Γ, (1) CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close Setting of the Control Problem (P) (P) min u∈M(Ω) J(u) = 1 2y − yd2 L2(Ω) + αuM(Ω), −∆y + c0y = u in Ω, y = 0 on Γ, (1) with c0 ∈ L∞(Ω) and c0 ≥ 0. We assume that α > 0, yd ∈ L2(Ω) and Ω is a bounded domain in Rn, n = 2 or 3, which is supposed to either be convex or have a C1,1 boundary Γ. The controls are taken in the space of regular Borel measures M(Ω). CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close Setting of the Control Problem (P) (P) min u∈M(Ω) J(u) = 1 2y − yd2 L2(Ω) + αuM(Ω), −∆y + c0y = u in Ω, y = 0 on Γ, (1) with c0 ∈ L∞(Ω) and c0 ≥ 0. We assume that α > 0, yd ∈ L2(Ω) and Ω is a bounded domain in Rn, n = 2 or 3, which is supposed to either be convex or have a C1,1 boundary Γ. The controls are taken in the space of regular Borel measures M(Ω). uM(Ω) = sup zC0(Ω)≤1 u, z = sup zC0(Ω)≤1
  • Ω
z(x) du = |u|(Ω) CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close Related Papers
  • C. Clason and K. Kunisch: “A duality-based approach to elliptic control
problems in non-reflexive Banach spaces”, ESAIM Control Optim. Calc. 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 L1 cost functional”. Submitted.
  • G. Stadler: “Elliptic optimal control problems with L1-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. CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close The State Equation Given a measure u ∈ M(Ω), we say that y is a solution to the state equation if
  • Ω
(−∆z + c0z)y dx =
  • Ω
z du for all z ∈ H2(Ω) ∩ H1 0(Ω) CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close The State Equation Given a measure u ∈ M(Ω), we say that y is a solution to the state equation if
  • Ω
(−∆z + c0z)y dx =
  • Ω
z du for all z ∈ H2(Ω) ∩ H1 0(Ω) It is well known that there exists a unique solution in this sense. More-
  • ver, y ∈ W 1,p
0 (Ω) for every 1 ≤ p < n n−1 and yW 1,p (Ω) ≤ CpuM(Ω) CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close Optimality Conditions THEOREM 1 The problem (P) has a unique solution ¯ u. CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close Optimality Conditions THEOREM 1 The problem (P) has a unique solution ¯
  • u. Moreover, if
¯ y denotes the associated state, and ¯ ϕ ∈ H2(Ω) ∩ H1 0(Ω) the adjoint state −∆ ¯ ϕ + c0 ¯ ϕ = ¯ y − yd in Ω ¯ ϕ =
  • n Γ
CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close Optimality Conditions THEOREM 1 The problem (P) has a unique solution ¯
  • u. Moreover, if
¯ y denotes the associated state, and ¯ ϕ ∈ H2(Ω) ∩ H1 0(Ω) the adjoint state −∆ ¯ ϕ + c0 ¯ ϕ = ¯ y − yd in Ω ¯ ϕ =
  • n Γ
then α¯ uM(Ω) +
  • Ω
¯ ϕ d¯ u = 0,
  • ¯
ϕC0(Ω) = α if ¯ u = 0, ¯ ϕC0(Ω) ≤ α if ¯ u = 0. CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close Sparsity If we consider the Jordan decomposition of ¯ u = ¯ u+ − ¯ u−, CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close Sparsity If we consider the Jordan decomposition of ¯ u = ¯ u+ − ¯ u−, then supp(¯ u+) ⊂ {x ∈ Ω : ¯ ϕ(x) = −α}, supp(¯ u−) ⊂ {x ∈ Ω : ¯ ϕ(x) = +α}. CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close Sparsity If we consider the Jordan decomposition of ¯ u = ¯ u+ − ¯ u−, then supp(¯ u+) ⊂ {x ∈ Ω : ¯ ϕ(x) = −α}, supp(¯ u−) ⊂ {x ∈ Ω : ¯ ϕ(x) = +α}. THEOREM 2 There exists ¯ α > 0 such that ¯ u = 0 for every α > ¯ α. CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close Sparsity If we consider the Jordan decomposition of ¯ u = ¯ u+ − ¯ u−, then supp(¯ u+) ⊂ {x ∈ Ω : ¯ ϕ(x) = −α}, supp(¯ u−) ⊂ {x ∈ Ω : ¯ ϕ(x) = +α}. THEOREM 2 There exists ¯ α > 0 such that ¯ u = 0 for every α > ¯ α. Proof. 1 2yα − yd2 L2(Ω) ≤ Jα(uα) CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close Sparsity If we consider the Jordan decomposition of ¯ u = ¯ u+ − ¯ u−, then supp(¯ u+) ⊂ {x ∈ Ω : ¯ ϕ(x) = −α}, supp(¯ u−) ⊂ {x ∈ Ω : ¯ ϕ(x) = +α}. THEOREM 2 There exists ¯ α > 0 such that ¯ u = 0 for every α > ¯ α. Proof. 1 2yα − yd2 L2(Ω) ≤ Jα(uα) ≤ Jα(0) CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close Sparsity If we consider the Jordan decomposition of ¯ u = ¯ u+ − ¯ u−, then supp(¯ u+) ⊂ {x ∈ Ω : ¯ ϕ(x) = −α}, supp(¯ u−) ⊂ {x ∈ Ω : ¯ ϕ(x) = +α}. THEOREM 2 There exists ¯ α > 0 such that ¯ u = 0 for every α > ¯ α. Proof. 1 2yα − yd2 L2(Ω) ≤ Jα(uα) ≤ Jα(0) = 1 2yd2 L2(Ω) CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close Sparsity If we consider the Jordan decomposition of ¯ u = ¯ u+ − ¯ u−, then supp(¯ u+) ⊂ {x ∈ Ω : ¯ ϕ(x) = −α}, supp(¯ u−) ⊂ {x ∈ Ω : ¯ ϕ(x) = +α}. THEOREM 2 There exists ¯ α > 0 such that ¯ u = 0 for every α > ¯ α. Proof. 1 2yα − yd2 L2(Ω) ≤ Jα(uα) ≤ Jα(0) = 1 2yd2 L2(Ω) ϕαC0(Ω) ≤ Cyα − ydL2(Ω) ≤ CydL2(Ω) CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close Sparsity If we consider the Jordan decomposition of ¯ u = ¯ u+ − ¯ u−, then supp(¯ u+) ⊂ {x ∈ Ω : ¯ ϕ(x) = −α}, supp(¯ u−) ⊂ {x ∈ Ω : ¯ ϕ(x) = +α}. THEOREM 2 There exists ¯ α > 0 such that ¯ u = 0 for every α > ¯ α. Proof. 1 2yα − yd2 L2(Ω) ≤ Jα(uα) ≤ Jα(0) = 1 2yd2 L2(Ω) ϕαC0(Ω) ≤ Cyα − ydL2(Ω) ≤ CydL2(Ω) = ¯ α CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close A Finite Element Approximation of (P)
  • Let us assume that Ω is convex and {Th}h>0 is a regular triangulation
  • f Ω satisfying an inverse assumption.
CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close A Finite Element Approximation of (P)
  • Let us assume that Ω is convex and {Th}h>0 is a regular triangulation
  • f Ω satisfying an inverse assumption. Ωh = ∪T∈ThT.
CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close A Finite Element Approximation of (P)
  • Let us assume that Ω is convex and {Th}h>0 is a regular triangulation
  • f Ω satisfying an inverse assumption. Ωh = ∪T∈ThT.
  • Discrete States:
Yh = {yh ∈ C(¯ Ω) | yh|T ∈ P1, for all T ∈ Th, and yh = 0 on ¯ Ω \ Ωh}, CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close A Finite Element Approximation of (P)
  • Let us assume that Ω is convex and {Th}h>0 is a regular triangulation
  • f Ω satisfying an inverse assumption. Ωh = ∪T∈ThT.
  • Discrete States:
Yh = {yh ∈ C(¯ Ω) | yh|T ∈ P1, for all T ∈ Th, and yh = 0 on ¯ Ω \ Ωh},
  • Discrete State Equation:
     Find yh ∈ Yh such that, for all zh ∈ Yh,
  • Ωh
[∇yh∇zh + c0yhzh] dx =
  • Ωh
zh du. CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close The Approximation (Ph) (Ph) min u∈M(Ω) Jh(uh) = 1 2yh − yd2 L2(Ωh) + αuM(Ω), where yh is the solution of the discrete state equation associated to u. CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close The Approximation (Ph) (Ph) min u∈M(Ω) Jh(uh) = 1 2yh − yd2 L2(Ωh) + αuM(Ω), where yh is the solution of the discrete state equation associated to u. Since we have not discretized the control space, this approach is re- lated to the variational discretization method introduced by Hinze. CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close The Approximation (Ph) (Ph) min u∈M(Ω) Jh(uh) = 1 2yh − yd2 L2(Ωh) + αuM(Ω), where yh is the solution of the discrete state equation associated to u. Since we have not discretized the control space, this approach is re- lated to the variational discretization method introduced by Hinze. We will show that among all the solutions to (Ph) there is a unique one which is a finite linear combination of Dirac measures concentrated in the interior vertices of the triangulation, leading to a simple numerical implementation. CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close Notation
  • {xj}N(h)
j=1 denote the interior nodes of the triangulation Th. CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close Notation
  • {xj}N(h)
j=1 denote the interior nodes of the triangulation Th.
  • {ej}N(h)
j=1 is the nodal basis of Yh: ej(xi) = δij. CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close Notation
  • {xj}N(h)
j=1 denote the interior nodes of the triangulation Th.
  • {ej}N(h)
j=1 is the nodal basis of Yh: ej(xi) = δij.
  • yh =
N(h)
  • j=1
yjej, where yj = yh(xj), 1 ≤ j ≤ N(h), ∀yh ∈ Yh. CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close Notation
  • {xj}N(h)
j=1 denote the interior nodes of the triangulation Th.
  • {ej}N(h)
j=1 is the nodal basis of Yh: ej(xi) = δij.
  • yh =
N(h)
  • j=1
yjej, where yj = yh(xj), 1 ≤ j ≤ N(h), ∀yh ∈ Yh.
  • Dh =
  uh ∈ M(Ω) : uh = N(h)
  • j=1
λjδxj, where {λj}N(h) j=1 ⊂ R    . CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close Notation
  • {xj}N(h)
j=1 denote the interior nodes of the triangulation Th.
  • {ej}N(h)
j=1 is the nodal basis of Yh: ej(xi) = δij.
  • yh =
N(h)
  • j=1
yjej, where yj = yh(xj), 1 ≤ j ≤ N(h), ∀yh ∈ Yh.
  • Dh =
  uh ∈ M(Ω) : uh = N(h)
  • j=1
λjδxj, where {λj}N(h) j=1 ⊂ R    .
  • Dh = Y ′
h, uh, yh = N(h)
  • j=1
λjyj ∀uh ∈ Dh, ∀yh ∈ Yh. CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close Two Linear Operators
  • Πh : C0(Ω) → Yh
Πhy = N(h)
  • j=1
y(xj)ej. CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close Two Linear Operators
  • Πh : C0(Ω) → Yh
Πhy = N(h)
  • j=1
y(xj)ej.
  • Λh : M(Ω) → Dh
Λhu = N(h)
  • j=1
u, ejδxj. CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close
  • For every u ∈ M(Ω) and every z ∈ C0(Ω) and zh ∈ Yh we have
u, zh = Λhu, zh and u, Πhz = Λhu, z CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close
  • For every u ∈ M(Ω) and every z ∈ C0(Ω) and zh ∈ Yh we have
u, zh = Λhu, zh and u, Πhz = Λhu, z
  • For every u ∈ M(Ω) we have
ΛhuM(Ω) ≤ uM(Ω) Λhu ∗ ⇀ u in M(Ω) and ΛhuM(Ω) → uM(Ω) CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close
  • For every u ∈ M(Ω) and every z ∈ C0(Ω) and zh ∈ Yh we have
u, zh = Λhu, zh and u, Πhz = Λhu, z
  • For every u ∈ M(Ω) we have
ΛhuM(Ω) ≤ uM(Ω) Λhu ∗ ⇀ u in M(Ω) and ΛhuM(Ω) → uM(Ω)
  • There exist a constant C > 0 such that for every u ∈ M(Ω)
u − ΛhuW −1,p(Ω) ≤ Ch1−n/p′uM(Ω), 1 < p < n n − 1 u − ΛhuW 1,∞ (Ω)∗ ≤ ChuM(Ω) CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close
  • For every u ∈ M(Ω) and every z ∈ C0(Ω) and zh ∈ Yh we have
u, zh = Λhu, zh and u, Πhz = Λhu, z
  • For every u ∈ M(Ω) we have
ΛhuM(Ω) ≤ uM(Ω) Λhu ∗ ⇀ u in M(Ω) and ΛhuM(Ω) → uM(Ω)
  • There exist a constant C > 0 such that for every u ∈ M(Ω)
u − ΛhuW −1,p(Ω) ≤ Ch1−n/p′uM(Ω), 1 < p < n n − 1 u − ΛhuW 1,∞ (Ω)∗ ≤ ChuM(Ω)
  • Given u ∈ M(Ω), let yh and ˜
yh be the discrete solutions associated to the controls u and Λhu, respectively, CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close
  • For every u ∈ M(Ω) and every z ∈ C0(Ω) and zh ∈ Yh we have
u, zh = Λhu, zh and u, Πhz = Λhu, z
  • For every u ∈ M(Ω) we have
ΛhuM(Ω) ≤ uM(Ω) Λhu ∗ ⇀ u in M(Ω) and ΛhuM(Ω) → uM(Ω)
  • There exist a constant C > 0 such that for every u ∈ M(Ω)
u − ΛhuW −1,p(Ω) ≤ Ch1−n/p′uM(Ω), 1 < p < n n − 1 u − ΛhuW 1,∞ (Ω)∗ ≤ ChuM(Ω)
  • Given u ∈ M(Ω), let yh and ˜
yh be the discrete solutions associated to the controls u and Λhu, respectively, then yh = ˜ yh. CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close The Solutions of (Ph) THEOREM 3 Problem (Ph) admits at least one solution. CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close The Solutions of (Ph) THEOREM 3 Problem (Ph) admits at least one solution. Among them there exists a unique one ¯ uh belonging to Dh. CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close The Solutions of (Ph) THEOREM 3 Problem (Ph) admits at least one solution. Among them there exists a unique one ¯ uh belonging to Dh. Moreover, any
  • ther solution ˜
uh ∈ M(Ω) of (Ph) satisfies that Λh˜ uh = ¯ uh. CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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  • f Cantabria
◭◭ ◮◮ ◭ ◮ Back Close The Solutions of (Ph) THEOREM 3 Problem (Ph) admits at least one solution. Among them there exists a unique one ¯ uh belonging to Dh. Moreover, any
  • ther solution ˜
uh ∈ M(Ω) of (Ph) satisfies that Λh˜ uh = ¯ uh. REMARK 1 The fact that (Ph) has exactly one solution in Dh is of practical interest. Indeed, recall that, as an element of Dh, ¯ uh has a unique representation of the form ¯ uh = N(h)
  • j=1
¯ λjδxj CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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13/20 UC University
  • f Cantabria
◭◭ ◮◮ ◭ ◮ Back Close The Solutions of (Ph) THEOREM 3 Problem (Ph) admits at least one solution. Among them there exists a unique one ¯ uh belonging to Dh. Moreover, any
  • ther solution ˜
uh ∈ M(Ω) of (Ph) satisfies that Λh˜ uh = ¯ uh. REMARK 1 The fact that (Ph) has exactly one solution in Dh is of practical interest. Indeed, recall that, as an element of Dh, ¯ uh has a unique representation of the form ¯ uh = N(h)
  • j=1
¯ λjδxj and ¯ uhM(Ω) = N(h)
  • j=1
|¯ λj| CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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13/20 UC University
  • f Cantabria
◭◭ ◮◮ ◭ ◮ Back Close The Solutions of (Ph) THEOREM 3 Problem (Ph) admits at least one solution. Among them there exists a unique one ¯ uh belonging to Dh. Moreover, any
  • ther solution ˜
uh ∈ M(Ω) of (Ph) satisfies that Λh˜ uh = ¯ uh. REMARK 1 The fact that (Ph) has exactly one solution in Dh is of practical interest. Indeed, recall that, as an element of Dh, ¯ uh has a unique representation of the form ¯ uh = N(h)
  • j=1
¯ λjδxj and ¯ uhM(Ω) = N(h)
  • j=1
|¯ λj| Then, the numerical computation of ¯ uh is reduced to the computation
  • f the coefficients {¯
λj}N(h) j=1 . CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close Convergence Analysis THEOREM 4 For every h > 0, let ¯ uh be the unique solution to (Ph) belonging to Dh and let ¯ u be the solution to (P). Then, the following convergence properties hold for h → 0: ¯ uh ∗ ⇀ ¯ u in M(Ω) CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close Convergence Analysis THEOREM 4 For every h > 0, let ¯ uh be the unique solution to (Ph) belonging to Dh and let ¯ u be the solution to (P). Then, the following convergence properties hold for h → 0: ¯ uh ∗ ⇀ ¯ u in M(Ω) ¯ uhM(Ω) → ¯ uM(Ω) CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close Convergence Analysis THEOREM 4 For every h > 0, let ¯ uh be the unique solution to (Ph) belonging to Dh and let ¯ u be the solution to (P). Then, the following convergence properties hold for h → 0: ¯ uh ∗ ⇀ ¯ u in M(Ω) ¯ uhM(Ω) → ¯ uM(Ω) ¯ y − ¯ yhL2(Ω) → 0 CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close Convergence Analysis THEOREM 4 For every h > 0, let ¯ uh be the unique solution to (Ph) belonging to Dh and let ¯ u be the solution to (P). Then, the following convergence properties hold for h → 0: ¯ uh ∗ ⇀ ¯ u in M(Ω) ¯ uhM(Ω) → ¯ uM(Ω) ¯ y − ¯ yhL2(Ω) → 0 Jh(¯ uh) → J(¯ u) where ¯ y and ¯ yh are the continuous and discrete states associated to ¯ u and ¯ uh, respectively. CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close Some Error Estimates Assumption: yd ∈ Lr(Ω) with r = 4 if n = 2 8 3 if n = 3 CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close Some Error Estimates Assumption: yd ∈ Lr(Ω) with r = 4 if n = 2 8 3 if n = 3 THEOREM 5 There exists a constant C > 0 independent of h such that |J(¯ u) − Jh(¯ uh)| ≤ Chκ where κ = 1 if n = 2 and κ = 1/2 if n = 3. CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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  • f Cantabria
◭◭ ◮◮ ◭ ◮ Back Close Some Error Estimates Assumption: yd ∈ Lr(Ω) with r = 4 if n = 2 8 3 if n = 3 THEOREM 5 There exists a constant C > 0 independent of h such that |J(¯ u) − Jh(¯ uh)| ≤ Chκ where κ = 1 if n = 2 and κ = 1/2 if n = 3. THEOREM 6 There exists a constant C > 0 independent of h such that ¯ y − ¯ yhL2(Ω) ≤ Ch κ 2 CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close Computational Results
  • Ωh = Ω = [−1, 1]2.
  • Uniform triangulation arising from N × N equidistributed nodes.
  • N = 128 (h ≈ 0.0157), c0 = 0, and α = 10−2.
  • yd = 10 exp(−50x2).
CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close Optimal Control ¯ uh CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close 10−2 10−1 100 0.5 1 1.5 2 2.5 3 3.5 4 ¯ α ≈ 0.187 α |uα|M(Ω) Dependence of uhM(Ω) on penalty parameter α CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close 10−2 10−1 10−2 10−1 h |Jh − Jh∗| O(h) Convergence order for the functionals CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011
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◭◭ ◮◮ ◭ ◮ Back Close 10−2 10−1 10−2 10−1 h yh − yh∗L2 O(h) Convergence order for the states CONTROL AND OPTIMIZATION OF PDES - GRAZ 2011