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Faculty of Mathematics Technische Universität München Sparse Optimal Control in Measure Spaces for Elliptic and Parabolic Systems Boris Vexler1 joint work with Karl Kunisch (KFU Graz) and Konstantin Pieper1 1 Faculty of Mathematics Chair of Optimal Control Technische Universität München June 2014 Boris Vexler Sparse Control Problems in Measure Spaces June 2014 1 Faculty of Mathematics Technische Universität München Outline 1. Motivation 2. Functional analytic setting 3. Optimality system and regularity 4. Discretization and error estimates 5. Parabolic case: Directional Sparsity Boris Vexler Sparse Control Problems in Measure Spaces June 2014 2 Faculty of Mathematics Technische Universität München Outline 1. Motivation 2. Functional analytic setting 3. Optimality system and regularity 4. Discretization and error estimates 5. Parabolic case: Directional Sparsity Boris Vexler Sparse Control Problems in Measure Spaces June 2014 3 Faculty of Mathematics Technische Universität München Motivation: Sparse control of PDEs I L1 -norm regularization leads to sparse solutions min Boris Vexler 1 kAx − bk22 + αkxk1 , 2 Sparse Control Problems in Measure Spaces x ∈ RN June 2014 4 Faculty of Mathematics Technische Universität München Motivation: Sparse control of PDEs I L1 -norm regularization leads to sparse solutions min 1 kAx − bk22 + αkxk1 , 2 x ∈ RN → Solution x̄ has typically only a handful ( N) of non-zero entries Boris Vexler Sparse Control Problems in Measure Spaces June 2014 4 Faculty of Mathematics Technische Universität München Motivation: Sparse control of PDEs I L1 -norm regularization leads to sparse solutions min 1 kAx − bk22 + αkxk1 , 2 x ∈ RN → Solution x̄ has typically only a handful ( N) of non-zero entries → For α ≥ α̂ Boris Vexler ⇒ x̄ = 0. Sparse Control Problems in Measure Spaces June 2014 4 Faculty of Mathematics Technische Universität München Motivation: Sparse control of PDEs I L1 -norm regularization leads to sparse solutions min 1 kAx − bk22 + αkxk1 , 2 x ∈ RN → Solution x̄ has typically only a handful ( N) of non-zero entries → For α ≥ α̂ I ⇒ x̄ = 0. Extension to optimal control problems with PDEs? Boris Vexler Sparse Control Problems in Measure Spaces June 2014 4 Faculty of Mathematics Technische Universität München Motivation: Sparse control of PDEs I L1 -norm regularization leads to sparse solutions min 1 kAx − bk22 + αkxk1 , 2 x ∈ RN → Solution x̄ has typically only a handful ( N) of non-zero entries → For α ≥ α̂ ⇒ x̄ = 0. I Extension to optimal control problems with PDEs? I Applications: I I optimal actuator placement, e.g., optimal distribution of light sources Point sourse identi cation Boris Vexler Sparse Control Problems in Measure Spaces June 2014 4 Faculty of Mathematics Technische Universität München Motivation: Sparse control of PDEs I L1 -norm regularization leads to sparse solutions min 1 kAx − bk22 + αkxk1 , 2 x ∈ RN → Solution x̄ has typically only a handful ( N) of non-zero entries → For α ≥ α̂ ⇒ x̄ = 0. I Extension to optimal control problems with PDEs? I Applications: I I I optimal actuator placement, e.g., optimal distribution of light sources Point sourse identi cation Right choice of function spaces? Boris Vexler Sparse Control Problems in Measure Spaces June 2014 4 Faculty of Mathematics Technische Universität München Motivation: Sparse control of PDEs I L1 -norm regularization leads to sparse solutions min 1 kAx − bk22 + αkxk1 , 2 x ∈ RN → Solution x̄ has typically only a handful ( N) of non-zero entries → For α ≥ α̂ ⇒ x̄ = 0. I Extension to optimal control problems with PDEs? I Applications: I I optimal actuator placement, e.g., optimal distribution of light sources Point sourse identi cation I Right choice of function spaces? I Discretization concept / error estimates / Optimization algorithms Boris Vexler Sparse Control Problems in Measure Spaces June 2014 4 Faculty of Mathematics Technische Universität München Motivation: Sparse control of PDEs Optimal control problem 1 Minimize J(q, u) = ku − ud k2L2 (Ω) + αkqkX , 2 ( −∆u = q in Ω, u=0 I q ∈ X, u ∈ V on ∂Ω, the standard setting X = L2 (Ω) does not lead to a sparse solution → for typical optimal control supp q̄ = Ω! Boris Vexler Sparse Control Problems in Measure Spaces June 2014 5 Faculty of Mathematics Technische Universität München Motivation: Sparse control of PDEs Optimal control problem 1 Minimize J(q, u) = ku − ud k2L2 (Ω) + αkqkX , 2 ( −∆u = q in Ω, u=0 I q ∈ X, u ∈ V on ∂Ω, the standard setting X = L2 (Ω) does not lead to a sparse solution → for typical optimal control supp q̄ = Ω! I Desired structure −∆u = N X qi δxi i=1 with unknown N and unknown positions xi of Diracs Boris Vexler Sparse Control Problems in Measure Spaces June 2014 5 Faculty of Mathematics Technische Universität München Motivation: Sparse control of parabolic PDEs Optimal control problem Minimize J(q, u) = 1 ku − ud k2L2 (I×Ω) + αkqkX , 2 ∂t u − ∆u = q in I × Ω, Boris Vexler q ∈ X, u ∈ V u(0) = u0 in Ω, Sparse Control Problems in Measure Spaces June 2014 6 Faculty of Mathematics Technische Universität München Motivation: Sparse control of parabolic PDEs Optimal control problem Minimize J(q, u) = 1 ku − ud k2L2 (I×Ω) + αkqkX , 2 ∂t u − ∆u = q in I × Ω, I q ∈ X, u ∈ V u(0) = u0 in Ω, Desired structure I ∂t u − ∆u = N X qi (t)δxi i=1 with unknown N and unknown (but xed) positions xi of Diracs Boris Vexler Sparse Control Problems in Measure Spaces June 2014 6 Faculty of Mathematics Technische Universität München Motivation: Sparse control of parabolic PDEs Optimal control problem Minimize J(q, u) = 1 ku − ud k2L2 (I×Ω) + αkqkX , 2 ∂t u − ∆u = q in I × Ω, I q ∈ X, u ∈ V u(0) = u0 in Ω, Desired structure I ∂t u − ∆u = N X qi (t)δxi i=1 I with unknown N and unknown (but xed) positions xi of Diracs Desired structure II N X ∂t u − ∆u = qi (t)δxi (t) i=1 with unknown N and unknown trajectories xi (t) of Diracs Boris Vexler Sparse Control Problems in Measure Spaces June 2014 6 Faculty of Mathematics Technische Universität München Outline 1. Motivation 2. Functional analytic setting 3. Optimality system and regularity 4. Discretization and error estimates 5. Parabolic case: Directional Sparsity Boris Vexler Sparse Control Problems in Measure Spaces June 2014 7 Faculty of Mathematics Technische Universität München Functional analytic setting Cost functional Minimize J(q, u) = 1 ku − ud k2L2 (Ω) + αkqkX , 2 q ∈ X, u ∈ V State equation −∆u = q u=0 Boris Vexler in Ω, on ∂Ω, Sparse Control Problems in Measure Spaces June 2014 8 Faculty of Mathematics Technische Universität München Functional analytic setting Cost functional Minimize J(q, u) = 1 ku − ud k2L2 (Ω) + αkqkX , 2 q ∈ X, u ∈ V State equation −∆u = q u=0 I in Ω, on ∂Ω, Choices for X? I X = L1 (Ω) I boundedness of a seq. {qn } in L1 (Ω) Boris Vexler ⇒ existence can not be guaranteed 6⇒ Sparse Control Problems in Measure Spaces qnk * q̄ June 2014 8 Faculty of Mathematics Technische Universität München Functional analytic setting Cost functional Minimize J(q, u) = 1 ku − ud k2L2 (Ω) + αkqkX , 2 q ∈ X, u ∈ V State equation −∆u = q u=0 I I in Ω, on ∂Ω, Choices for X? I X = L1 (Ω) I boundedness of a seq. {qn } in L1 (Ω) ⇒ existence can not be guaranteed 6⇒ qnk * q̄ choose X = M(Ω), the space of regular Borel measures on Ω Boris Vexler Sparse Control Problems in Measure Spaces June 2014 8 Faculty of Mathematics Technische Universität München Functional analytic setting Cost functional Minimize J(q, u) = 1 ku − ud k2L2 (Ω) + αkqkX , 2 q ∈ X, u ∈ V State equation −∆u = q u=0 I I in Ω, on ∂Ω, Choices for X? I X = L1 (Ω) I boundedness of a seq. {qn } in L1 (Ω) ⇒ existence can not be guaranteed 6⇒ qnk * q̄ choose X = M(Ω), the space of regular Borel measures on Ω I C. Clason and K. Kunisch A duality based approach to control problems in non-re exive Banach spaces ESAIM COCV 17, pp. 243-266, 2011 Boris Vexler Sparse Control Problems in Measure Spaces June 2014 8 Faculty of Mathematics Technische Universität München Example I: Sparse controls Minimize J(q, u) = 1 ku − ud k2L2 (Ω0 ) + αkqkM(Ωc ) 2 subject to −∆u − b · ∇u = q in Ω, (a) ud on Ωo Boris Vexler (b) optimal state ū u = 0 on ∂Ω (c) optimal control q̄ on Ωc Sparse Control Problems in Measure Spaces June 2014 9 Faculty of Mathematics Technische Universität München Example II: Inverse point source reconstruction I state equation: ∂t u − ∆u − b · ∇u = q & boundary and initial cond. Γin b(x) Ωc x0 Ω b(x) Γout Boris Vexler Sparse Control Problems in Measure Spaces June 2014 10 Faculty of Mathematics Technische Universität München Literature I Measure valued controls in elliptic equations I G. Buttazzo, N. Varchon and H. Zoubairi Optimal measures for elliptic problems Ann. Mat. Pura Appl. 185, pp. 207 221, 2006. I C. Clason and K. Kunisch A duality based approach to control problems in non-re exive Banach spaces ESAIM COCV 17, pp. 243-266, 2011 I E. Casas, C. Clason and K. Kunisch Approximation of elliptic control problems in measure spaces with sparse solutions SIAM J. Control Optim. 50, pp. 1735 1752, 2012 I K. Pieper and B. Vexler A priori error analysis for discretization of sparse elliptic optimal control problems in measure spase SIAM J. Control Optim. 51, pp. 2788-2808, 2013 I E. Casas and K. Kunisch Optimal control of semilinear elliptic equations in measure spaces SIAM J. Control Optim. 52 , pp. 339-364, 2013 I Measure valued controls in parabolic equations I E. Casas and E. Zuazua Spike controls for elliptic and parabolic PDE Systems and Control letters 63, pp. 311-318, 2013 I E. Casas, C. Clason and K. Kunisch Parabolic control problems in measure spaces with sparse solutions SIAM J. Control Optim. 51, pp. 28-63, 2013 I K. Kunisch, K. Pieper and B. Vexler Measure valued directional sparsity for parabolic optimal control problems SIAM J. Control Optim., submitted Boris Vexler Sparse Control Problems in Measure Spaces June 2014 11 Faculty of Mathematics Technische Universität München Functional analytic setting I Choose X = M(Ω), the space of regular Borel measures on Ω Boris Vexler Sparse Control Problems in Measure Spaces June 2014 12 Faculty of Mathematics Technische Universität München Functional analytic setting I I Choose X = M(Ω), the space of regular Borel measures on Ω Identify M(Ω) = (C0 (Ω))∗ with Z hq, ϕi = ϕ dq, q ∈ M(Ω), ϕ ∈ C0 (Ω) Ω Boris Vexler Sparse Control Problems in Measure Spaces June 2014 12 Faculty of Mathematics Technische Universität München Functional analytic setting I I Choose X = M(Ω), the space of regular Borel measures on Ω Identify M(Ω) = (C0 (Ω))∗ with Z hq, ϕi = ϕ dq, q ∈ M(Ω), ϕ ∈ C0 (Ω) Ω I Norm for q ∈ M(Ω) kqkM(Ω) = sup hq, ϕi ϕ∈C0 (Ω) kϕkC0 (Ω) ≤1 Boris Vexler Sparse Control Problems in Measure Spaces June 2014 12 Faculty of Mathematics Technische Universität München Functional analytic setting I I Choose X = M(Ω), the space of regular Borel measures on Ω Identify M(Ω) = (C0 (Ω))∗ with Z hq, ϕi = ϕ dq, q ∈ M(Ω), ϕ ∈ C0 (Ω) Ω I Norm for q ∈ M(Ω) kqkM(Ω) = sup hq, ϕi ϕ∈C0 (Ω) kϕkC0 (Ω) ≤1 → Other possibility X = L1 (Ω) with an additional L2 -regularization and/or control constraints (G. Stadler, G. Wachsmuth, R. Herzog) Boris Vexler Sparse Control Problems in Measure Spaces June 2014 12 Faculty of Mathematics Technische Universität München Functional analytic setting: State equation State equation: very weak formulation for q ∈ M(Ω) u ∈ L2 (Ω) : (u, −∆ϕ) = hq, ϕi Boris Vexler ∀ϕ ∈ H2 (Ω) ∩ H10 (Ω) Sparse Control Problems in Measure Spaces June 2014 13 Faculty of Mathematics Technische Universität München Functional analytic setting: State equation State equation: very weak formulation for q ∈ M(Ω) u ∈ L2 (Ω) : (u, −∆ϕ) = hq, ϕi ∀ϕ ∈ H2 (Ω) ∩ H10 (Ω) Regularity u ∈ W01,sε (Ω), Boris Vexler sε = d − ε, d−1 for all 0 < ε < Sparse Control Problems in Measure Spaces 1 d−1 June 2014 13 Faculty of Mathematics Technische Universität München Functional analytic setting: State equation State equation: very weak formulation for q ∈ M(Ω) u ∈ L2 (Ω) : (u, −∆ϕ) = hq, ϕi ∀ϕ ∈ H2 (Ω) ∩ H10 (Ω) Regularity u ∈ W01,sε (Ω), sε = d − ε, d−1 for all 0 < ε < 1 d−1 Lemma: Stability estimate k∇ukLsε (Ω) ≤ Boris Vexler c kqkM(Ω) ε Sparse Control Problems in Measure Spaces June 2014 13 Faculty of Mathematics Technische Universität München Functional analytic setting: State equation State equation: very weak formulation for q ∈ M(Ω) u ∈ L2 (Ω) : (u, −∆ϕ) = hq, ϕi ∀ϕ ∈ H2 (Ω) ∩ H10 (Ω) Regularity u ∈ W01,sε (Ω), sε = d − ε, d−1 for all 0 < ε < 1 d−1 Lemma: Stability estimate k∇ukLsε (Ω) ≤ c kqkM(Ω) ε Weak formulation for q ∈ M(Ω) u ∈ W01,sε (Ω) : (∇u, ∇ϕ) = hq, ϕi Boris Vexler 1,s0ε ∀ϕ ∈ W0 Sparse Control Problems in Measure Spaces (Ω) ,→ C0 (Ω) June 2014 13 Faculty of Mathematics Technische Universität München Functional analytic setting Optimal control problem Minimize J(q, u) = 1 ku − ud k2L2 (Ω ) + αkqkM(Ω ) , 2 q ∈ M(Ω), subject to u ∈ W01,sε (Ω) : (∇u, ∇ϕ) = hq, ϕi Boris Vexler Sparse Control Problems in Measure Spaces 1,s0ε ∀ϕ ∈ W0 (Ω) June 2014 14 Faculty of Mathematics Technische Universität München Functional analytic setting Optimal control problem Minimize J(q, u) = 1 ku − ud k2L2 (Ω ) + αkqkM(Ω ) , 2 q ∈ M(Ω), subject to u ∈ W01,sε (Ω) : (∇u, ∇ϕ) = hq, ϕi I 1,s0ε ∀ϕ ∈ W0 (Ω) Existence of (q̄, ū) I I minimizing sequence {qn }, bounded in M(Ω) = (C0 (Ω))∗ Banach-Alaoglu theorem: {qnk } converging weakly∗ to q̄ ∈ M(Ω) Boris Vexler Sparse Control Problems in Measure Spaces June 2014 14 Faculty of Mathematics Technische Universität München Functional analytic setting Optimal control problem Minimize J(q, u) = 1 ku − ud k2L2 (Ω ) + αkqkM(Ω ) , 2 q ∈ M(Ω), subject to u ∈ W01,sε (Ω) : (∇u, ∇ϕ) = hq, ϕi I (Ω) Existence of (q̄, ū) I I I 1,s0ε ∀ϕ ∈ W0 minimizing sequence {qn }, bounded in M(Ω) = (C0 (Ω))∗ Banach-Alaoglu theorem: {qnk } converging weakly∗ to q̄ ∈ M(Ω) Uniqueness of (q̄, ū) I unique ū, injectivity of the solution operator ⇒ unique q̄ Boris Vexler Sparse Control Problems in Measure Spaces June 2014 14 Faculty of Mathematics Technische Universität München Functional analytic setting Optimal control problem Minimize J(q, u) = 1 ku − ud k2L2 (Ωo ) + αkqkM(Ωc ) , 2 q ∈ M(Ω), subject to u ∈ W01,sε (Ω) : (∇u, ∇ϕ) = hq, ϕi I (Ω) Existence of (q̄, ū) I I I 1,s0ε ∀ϕ ∈ W0 minimizing sequence {qn }, bounded in M(Ω) = (C0 (Ω))∗ Banach-Alaoglu theorem: {qnk } converging weakly∗ to q̄ ∈ M(Ω) Uniqueness of (q̄, ū) I I unique ū, injectivity of the solution operator ⇒ unique q̄ disjoint control and observation domains ⇒ multiple solutions! Boris Vexler Sparse Control Problems in Measure Spaces June 2014 14 Faculty of Mathematics Technische Universität München Outline 1. Motivation 2. Functional analytic setting 3. Optimality system and regularity 4. Discretization and error estimates 5. Parabolic case: Directional Sparsity Boris Vexler Sparse Control Problems in Measure Spaces June 2014 15 Faculty of Mathematics Technische Universität München Optimality conditions Reduced formulation Minimize j(q) = f(q) + αkqkM(Ω) , Boris Vexler with f(q) = Sparse Control Problems in Measure Spaces 1 ku(q) − ud k2L2 (Ω) 2 June 2014 16 Faculty of Mathematics Technische Universität München Optimality conditions Reduced formulation Minimize j(q) = f(q) + αkqkM(Ω) , with f(q) = 1 ku(q) − ud k2L2 (Ω) 2 Optimality condition 0 ∈ ∂j(q̄) Boris Vexler ⇔ α kqkM(Ω) − kq̄kM(Ω) ≥ −f 0 (q̄)(q−q̄) Sparse Control Problems in Measure Spaces ∀q ∈ M(Ω) June 2014 16 Faculty of Mathematics Technische Universität München Optimality conditions Reduced formulation Minimize j(q) = f(q) + αkqkM(Ω) , with f(q) = 1 ku(q) − ud k2L2 (Ω) 2 Optimality condition 0 ∈ ∂j(q̄) ⇔ α kqkM(Ω) − kq̄kM(Ω) ≥ −f 0 (q̄)(q−q̄) ∀q ∈ M(Ω) Optimality condition (u(q) − ū, ū − ud ) + α kqkM(Ω) − kq̄kM(Ω) ≥ 0 ∀q ∈ M(Ω) Boris Vexler Sparse Control Problems in Measure Spaces June 2014 16 Faculty of Mathematics Technische Universität München Optimality system State equation for ū ∈ W01,s (Ω) −∆ū = q̄ ū = 0 in Ω, on ∂Ω, Adjoint equation for z̄ ∈ H2 (Ω) ∩ H10 (Ω) −∆z̄ = ū − ud z̄ = 0 in Ω, on ∂Ω, Optimality condition for q̄ ∈ M(Ω) −hq − q̄, z̄i + αkq̄kM(Ω) ≤ αkqkM(Ω) Boris Vexler Sparse Control Problems in Measure Spaces ∀q ∈ M(Ω) June 2014 17 Faculty of Mathematics Technische Universität München Consequences from the optimality system I kz̄kC0 (Ω) ≤ α Boris Vexler Sparse Control Problems in Measure Spaces June 2014 18 Faculty of Mathematics Technische Universität München Consequences from the optimality system I kz̄kC0 (Ω) ≤ α I Jordan decomposition q̄ = q̄+ − q̄− supp q̄+ ⊂ { x ∈ Ω | z̄(x) = −α } , supp q̄− ⊂ { x ∈ Ω | z̄(x) = α } → sparsity! Boris Vexler Sparse Control Problems in Measure Spaces June 2014 18 Faculty of Mathematics Technische Universität München Consequences from the optimality system I kz̄kC0 (Ω) ≤ α I Jordan decomposition q̄ = q̄+ − q̄− supp q̄+ ⊂ { x ∈ Ω | z̄(x) = −α } , supp q̄− ⊂ { x ∈ Ω | z̄(x) = α } → sparsity! I z̄ ∈ H2 (Ω) ∩ H10 (Ω) ,→ C0,β (Ω̄) supp q̄ ⊂ Ω η = { x ∈ Ω | dist (x, ∂Ω) > η } , Boris Vexler Sparse Control Problems in Measure Spaces η>0 June 2014 18 Faculty of Mathematics Technische Universität München Consequences from the optimality system I kz̄kC0 (Ω) ≤ α I Jordan decomposition q̄ = q̄+ − q̄− supp q̄+ ⊂ { x ∈ Ω | z̄(x) = −α } , supp q̄− ⊂ { x ∈ Ω | z̄(x) = α } → sparsity! I z̄ ∈ H2 (Ω) ∩ H10 (Ω) ,→ C0,β (Ω̄) supp q̄ ⊂ Ω η = { x ∈ Ω | dist (x, ∂Ω) > η } , I η>0 dist (supp q̄+ , supp q̄− ) > η Boris Vexler Sparse Control Problems in Measure Spaces June 2014 18 Faculty of Mathematics Technische Universität München Improved regularity I examples with q̄ = δx0 (ud behaves like a Green's function) Boris Vexler Sparse Control Problems in Measure Spaces June 2014 19 Faculty of Mathematics Technische Universität München Improved regularity I examples with q̄ = δx0 (ud behaves like a Green's function) I numerical observation: if Ωo = Ωc = Ω ud ∈ L∞ (Ω) Boris Vexler ⇒ ū ∈ L∞ (Ω) Sparse Control Problems in Measure Spaces June 2014 19 Faculty of Mathematics Technische Universität München Improved regularity I examples with q̄ = δx0 (ud behaves like a Green's function) I numerical observation: if Ωo = Ωc = Ω ud ∈ L∞ (Ω) I ⇒ ū ∈ L∞ (Ω) proof? Boris Vexler Sparse Control Problems in Measure Spaces June 2014 19 Faculty of Mathematics Technische Universität München Improved regularity I examples with q̄ = δx0 (ud behaves like a Green's function) I numerical observation: if Ωo = Ωc = Ω ud ∈ L∞ (Ω) I proof? I consequences: ⇒ ū ∈ L∞ (Ω) I excludes Diracs I ū ∈ H10 (Ω) ∩ L∞ (Ω) I q̄ ∈ H−1 (Ω) ∩ M(Ω) I typical solutions: line measures in 2d and surface measures in 3d Boris Vexler Sparse Control Problems in Measure Spaces June 2014 19 Faculty of Mathematics Technische Universität München Improved regularity I examples with q̄ = δx0 (ud behaves like a Green's function) I numerical observation: if Ωo = Ωc = Ω ud ∈ L∞ (Ω) I proof? I consequences: I ⇒ ū ∈ L∞ (Ω) I excludes Diracs I ū ∈ H10 (Ω) ∩ L∞ (Ω) I q̄ ∈ H−1 (Ω) ∩ M(Ω) I typical solutions: line measures in 2d and surface measures in 3d for Ω0 ∩ Ωc = ∅ typical solution is a lin. combination of Diracs Boris Vexler Sparse Control Problems in Measure Spaces June 2014 19 Faculty of Mathematics Technische Universität München Improved regularity Theorem (K. Pieper and B.V., SICON 2013) Let Ωo = Ωc = Ω and ud ∈ L∞ (Ω). Then there holds (a) ū ∈ L∞ (Ω) with kūkL∞ (Ω) ≤ kud kL∞ (Ω) (b) ū ∈ H10 (Ω) ∩ L∞ (Ω) (c) q̄ ∈ H−1 (Ω) ∩ M(Ω) Boris Vexler Sparse Control Problems in Measure Spaces June 2014 20 Faculty of Mathematics Technische Universität München Improved regularity Theorem (K. Pieper and B.V., SICON 2013) Let Ωo = Ωc = Ω and ud ∈ L∞ (Ω). Then there holds (a) ū ∈ L∞ (Ω) with kūkL∞ (Ω) ≤ kud kL∞ (Ω) (b) ū ∈ H10 (Ω) ∩ L∞ (Ω) (c) q̄ ∈ H−1 (Ω) ∩ M(Ω) I is not true for Ωc ∩ Ωo = ∅ Boris Vexler Sparse Control Problems in Measure Spaces June 2014 20 Faculty of Mathematics Technische Universität München Improved regularity Theorem (K. Pieper and B.V., SICON 2013) Let Ωo = Ωc = Ω and ud ∈ L∞ (Ω). Then there holds (a) ū ∈ L∞ (Ω) with kūkL∞ (Ω) ≤ kud kL∞ (Ω) (b) ū ∈ H10 (Ω) ∩ L∞ (Ω) (c) q̄ ∈ H−1 (Ω) ∩ M(Ω) I I is not true for Ωc ∩ Ωo = ∅ Proof uses I I I potential theory maximum principle (with measures as rhs) optimality system Boris Vexler Sparse Control Problems in Measure Spaces June 2014 20 Faculty of Mathematics Technische Universität München Outline 1. Motivation 2. Functional analytic setting 3. Optimality system and regularity 4. Discretization and error estimates 5. Parabolic case: Directional Sparsity Boris Vexler Sparse Control Problems in Measure Spaces June 2014 21 Faculty of Mathematics Technische Universität München FEM-Discretization and error estimates I I Discretization concept following Casas/Clason/Kunisch [CCK2011] E. Casas, C. Clason and K. Kunisch Approximation of elliptic control problems in measure spaces with sparse solutions SIAM J. Control Optim. 50 (2012), pp. 1735-1752 Boris Vexler Sparse Control Problems in Measure Spaces June 2014 22 Faculty of Mathematics Technische Universität München FEM-Discretization and error estimates I I I Discretization concept following Casas/Clason/Kunisch [CCK2011] error estimates from [CCK2012] d kū − ūh kL2 (Ω) = O h1− 4 E. Casas, C. Clason and K. Kunisch Approximation of elliptic control problems in measure spaces with sparse solutions SIAM J. Control Optim. 50 (2012), pp. 1735-1752 Boris Vexler Sparse Control Problems in Measure Spaces June 2014 22 Faculty of Mathematics Technische Universität München FEM-Discretization and error estimates I I I Discretization concept following Casas/Clason/Kunisch [CCK2011] error estimates from [CCK2012] d kū − ūh kL2 (Ω) = O h1− 4 estimate in [PV2013] in the general case d kū − ūh kL2 (Ω) = O h2− 2 |ln h| I E. Casas, C. Clason and K. Kunisch Approximation of elliptic control problems in measure spaces with sparse solutions SIAM J. Control Optim. 50 (2012), pp. 1735-1752 I K. Pieper and B. Vexler A priori error analysis for discretization of sparse elliptic optimal control problems in measure spase SIAM J. Control Optim. 51, pp. 2788-2808, 2013 Boris Vexler Sparse Control Problems in Measure Spaces June 2014 22 Faculty of Mathematics Technische Universität München FEM-Discretization and error estimates I I Discretization concept following Casas/Clason/Kunisch [CCK2011] error estimates from [CCK2012] d kū − ūh kL2 (Ω) = O h1− 4 I estimate in [PV2013] in the general case d kū − ūh kL2 (Ω) = O h2− 2 |ln h| I estimate in [PV2013] for ud ∈ L∞ (Ω) and Ωo = Ωc = Ω (improved regularity) kū − ūh kL2 (Ω) = O (h |ln h|) I E. Casas, C. Clason and K. Kunisch Approximation of elliptic control problems in measure spaces with sparse solutions SIAM J. Control Optim. 50 (2012), pp. 1735-1752 I K. Pieper and B. Vexler A priori error analysis for discretization of sparse elliptic optimal control problems in measure spase SIAM J. Control Optim. 51, pp. 2788-2808, 2013 Boris Vexler Sparse Control Problems in Measure Spaces June 2014 22 Faculty of Mathematics Technische Universität München Discretization concept I discretize the state variable with linear nite elements, ( rst) don't discretize the control Boris Vexler Sparse Control Problems in Measure Spaces June 2014 23 Faculty of Mathematics Technische Universität München Discretization concept I discretize the state variable with linear nite elements, ( rst) don't discretize the control Discretized optimal control problem Minimize J(qh , uh ) = 1 ku − ud k2L2 (Ω) + αkqh kM(Ω) , 2 h qh ∈ M(Ω), subject to uh ∈ Vh : (∇uh , ∇ϕh ) = hqh , ϕh i Boris Vexler Sparse Control Problems in Measure Spaces ∀ϕh ∈ Vh June 2014 23 Faculty of Mathematics Technische Universität München Discretization concept I discretize the state variable with linear nite elements, ( rst) don't discretize the control Discretized optimal control problem Minimize J(qh , uh ) = 1 ku − ud k2L2 (Ω) + αkqh kM(Ω) , 2 h qh ∈ M(Ω), subject to uh ∈ Vh : (∇uh , ∇ϕh ) = hqh , ϕh i I ∀ϕh ∈ Vh existence as on the continuous level, ūh is unique Boris Vexler Sparse Control Problems in Measure Spaces June 2014 23 Faculty of Mathematics Technische Universität München Discretization concept I discretize the state variable with linear nite elements, ( rst) don't discretize the control Discretized optimal control problem Minimize J(qh , uh ) = 1 ku − ud k2L2 (Ω) + αkqh kM(Ω) , 2 h qh ∈ M(Ω), subject to uh ∈ Vh : (∇uh , ∇ϕh ) = hqh , ϕh i I I ∀ϕh ∈ Vh existence as on the continuous level, ūh is unique discrete solution operator Sh : q 7→ uh (q) is not injective Boris Vexler Sparse Control Problems in Measure Spaces June 2014 23 Faculty of Mathematics Technische Universität München Discretization concept I discretize the state variable with linear nite elements, ( rst) don't discretize the control Discretized optimal control problem Minimize J(qh , uh ) = 1 ku − ud k2L2 (Ω) + αkqh kM(Ω) , 2 h qh ∈ M(Ω), subject to uh ∈ Vh : (∇uh , ∇ϕh ) = hqh , ϕh i I I I ∀ϕh ∈ Vh existence as on the continuous level, ūh is unique discrete solution operator Sh : q 7→ uh (q) is not injective discrete optimal control is not unique! Boris Vexler Sparse Control Problems in Measure Spaces June 2014 23 Faculty of Mathematics Technische Universität München Discretization concept I with interior nodes {xi } of the mesh Th and basis functions ϕi ∈ Vh X Mh = { qh ∈ M(Ω) | qh = βi δxi , βi ∈ R } ⊂ M(Ω) i Boris Vexler Sparse Control Problems in Measure Spaces June 2014 24 Faculty of Mathematics Technische Universität München Discretization concept I with interior nodes {xi } of the mesh Th and basis functions ϕi ∈ Vh X Mh = { qh ∈ M(Ω) | qh = βi δxi , βi ∈ R } ⊂ M(Ω) i I Λh : M(Ω) → Mh Λh q = X hq, ϕi iδxi i Boris Vexler Sparse Control Problems in Measure Spaces June 2014 24 Faculty of Mathematics Technische Universität München Discretization concept I with interior nodes {xi } of the mesh Th and basis functions ϕi ∈ Vh X Mh = { qh ∈ M(Ω) | qh = βi δxi , βi ∈ R } ⊂ M(Ω) i I Λh : M(Ω) → Mh Λh q = X hq, ϕi iδxi i Theorem (Casas/Clason/Kunisch) There exists a unique solution q̄h ∈ Mh . For any other solution q̃h ∈ M(Ω) there holds Λh q̃h = q̄h . Moreover ∗ q̄h * q̄ in M(Ω) Boris Vexler and kq̄h kM(Ω) → kq̄kM(Ω) Sparse Control Problems in Measure Spaces June 2014 24 Faculty of Mathematics Technische Universität München Discrete optimality system State equation for ūh ∈ Vh ūh ∈ Vh : (∇ūh , ∇ϕh ) = hq̄h , ϕh i ∀ϕh ∈ Vh Adjoint equation for z̄h ∈ Vh z̄h ∈ Vh : (∇ϕh , ∇z̄h ) = (ūh − ud , ϕh ) ∀ϕh ∈ Vh Optimality condition for q̄h ∈ Mh −hq − q̄h , z̄h i + αkq̄h kM(Ω) ≤ αkqkM(Ω) Boris Vexler Sparse Control Problems in Measure Spaces ∀q ∈ M(Ω) June 2014 25 Faculty of Mathematics Technische Universität München Error estimates for the state equation I estimate in L2 (Ω)-norm d ku(q) − uh (q)kL2 (Ω) ≤ ch2− 2 kqkM(Ω) → does not lead to optimal estimates Boris Vexler Sparse Control Problems in Measure Spaces June 2014 26 Faculty of Mathematics Technische Universität München Error estimates for the state equation I estimate in L2 (Ω)-norm d ku(q) − uh (q)kL2 (Ω) ≤ ch2− 2 kqkM(Ω) → does not lead to optimal estimates I estimate in Lp (Ω)-norm, 1 < p < ku(q) − uh (q)kLp (Ω) ≤ cp h Boris Vexler d d−2 2− pd0 kqkM(Ω) , Sparse Control Problems in Measure Spaces 1 1 + 0 =1 p p June 2014 26 Faculty of Mathematics Technische Universität München Error estimates for the state equation I estimate in L2 (Ω)-norm d ku(q) − uh (q)kL2 (Ω) ≤ ch2− 2 kqkM(Ω) → does not lead to optimal estimates I estimate in Lp (Ω)-norm, 1 < p < ku(q) − uh (q)kLp (Ω) ≤ cp h I d d−2 2− pd0 kqkM(Ω) , 1 1 + 0 =1 p p estimate in L1 (Ω)-norm ku(q) − uh (q)kL1 (Ω) ≤ ch2 |ln h|2 kqkM(Ω) Boris Vexler Sparse Control Problems in Measure Spaces June 2014 26 Faculty of Mathematics Technische Universität München Error estimates for the state equation I estimate in L2 (Ω)-norm d ku(q) − uh (q)kL2 (Ω) ≤ ch2− 2 kqkM(Ω) → does not lead to optimal estimates I estimate in Lp (Ω)-norm, 1 < p < ku(q) − uh (q)kLp (Ω) ≤ cp h I d d−2 2− pd0 kqkM(Ω) , 1 1 + 0 =1 p p estimate in L1 (Ω)-norm ku(q) − uh (q)kL1 (Ω) ≤ ch2 |ln h|2 kqkM(Ω) I proof uses a duality argument and appropriate L∞ -estimates Boris Vexler Sparse Control Problems in Measure Spaces June 2014 26 Faculty of Mathematics Technische Universität München Error estimates for the optimal control problem I general assumption on ud ud ∈ L∞ (Ω) in 2d and ud ∈ L3 (Ω) in 3d I general regularity: q̄ ∈ M(Ω) and ū ∈ W01,s (Ω) Boris Vexler Sparse Control Problems in Measure Spaces June 2014 27 Faculty of Mathematics Technische Universität München Error estimates for the optimal control problem I general assumption on ud ud ∈ L∞ (Ω) in 2d and ud ∈ L3 (Ω) in 3d I general regularity: q̄ ∈ M(Ω) and ū ∈ W01,s (Ω) Theorem (K. Pieper and B.V.) Let (q̄, ū) be the optimal solution and (q̄h , ūh ) ∈ Mh × Vh the discrete optimal solution. There holds d kū − ūh kL2 (Ω) ≤ ch2− 2 |ln h| Boris Vexler Sparse Control Problems in Measure Spaces June 2014 27 Faculty of Mathematics Technische Universität München Error estimates for the optimal control problem I general assumption on ud ud ∈ L∞ (Ω) in 2d and ud ∈ L3 (Ω) in 3d I general regularity: q̄ ∈ M(Ω) and ū ∈ W01,s (Ω) Theorem (K. Pieper and B.V.) Let (q̄, ū) be the optimal solution and (q̄h , ūh ) ∈ Mh × Vh the discrete optimal solution. There holds d kū − ūh kL2 (Ω) ≤ ch2− 2 |ln h| I Corollary: d kq̄ − q̄h kH−2 (Ω) ≤ ch2− 2 |ln h| Boris Vexler Sparse Control Problems in Measure Spaces June 2014 27 Faculty of Mathematics Technische Universität München Error estimates for the optimal control problem I general assumption on ud ud ∈ L∞ (Ω) in 2d and ud ∈ L3 (Ω) in 3d I general regularity: q̄ ∈ M(Ω) and ū ∈ W01,s (Ω) Theorem (K. Pieper and B.V.) Let (q̄, ū) be the optimal solution and (q̄h , ūh ) ∈ Mh × Vh the discrete optimal solution. There holds d kū − ūh kL2 (Ω) ≤ ch2− 2 |ln h| I Corollary: d kq̄ − q̄h kH−2 (Ω) ≤ ch2− 2 |ln h| I in general one can not expect kq̄ − q̄h kM(Ω) → 0. Boris Vexler Sparse Control Problems in Measure Spaces June 2014 27 Faculty of Mathematics Technische Universität München Error estimate for improved regularity I assume ud ∈ L∞ (Ω) and Ωo = Ωc = Ω Boris Vexler Sparse Control Problems in Measure Spaces June 2014 28 Faculty of Mathematics Technische Universität München Error estimate for improved regularity I assume ud ∈ L∞ (Ω) and Ωo = Ωc = Ω I improved regularity: q̄ ∈ M(Ω) ∩ H−1 (Ω) Boris Vexler and ū ∈ L∞ (Ω) ∩ H10 (Ω) Sparse Control Problems in Measure Spaces June 2014 28 Faculty of Mathematics Technische Universität München Error estimate for improved regularity I assume ud ∈ L∞ (Ω) and Ωo = Ωc = Ω I improved regularity: q̄ ∈ M(Ω) ∩ H−1 (Ω) and ū ∈ L∞ (Ω) ∩ H10 (Ω) Theorem (K. Pieper and B.V.) Let (q̄, ū) be the optimal solution and (q̄h , ūh ) ∈ Mh × Vh the discrete optimal solution. There holds kū − ūh kL2 (Ω) ≤ ch|ln h| Boris Vexler Sparse Control Problems in Measure Spaces June 2014 28 Faculty of Mathematics Technische Universität München Error estimate for improved regularity I assume ud ∈ L∞ (Ω) and Ωo = Ωc = Ω I improved regularity: q̄ ∈ M(Ω) ∩ H−1 (Ω) and ū ∈ L∞ (Ω) ∩ H10 (Ω) Theorem (K. Pieper and B.V.) Let (q̄, ū) be the optimal solution and (q̄h , ūh ) ∈ Mh × Vh the discrete optimal solution. There holds kū − ūh kL2 (Ω) ≤ ch|ln h| I proof uses optimality systems, z̄ ∈ W 2,p (Ω), p < ∞ I applies L∞ error estimates for z̄ and L2 estimates for ū Boris Vexler Sparse Control Problems in Measure Spaces June 2014 28 Faculty of Mathematics Technische Universität München Numerical results: Dirac example (I) I construct a radially symmetric example on Ω = B1 (0) q̄ = δ0 , I ū = (−∆)−1 δ0 , z̄ with z̄(0) = −α set ud = ū + ∆z̄ (d) State 2d Boris Vexler (e) Adjoint 2d Sparse Control Problems in Measure Spaces (f) Control 2d June 2014 29 Faculty of Mathematics Technische Universität München Numerical results: Dirac example (II) |J(q̄, ū) − J(q̄h , ūh )| kū − ūh kL2 (Ω) 2 ε 2 kq̄h kL2 100 10−1 h −2 |J(q̄, ū) − J(q̄h , ūh )| kū − ūh kL2 (Ω) 2 ε 2 kq̄h kL2 100 10 10 −4 h −2 10 10−3 10−6 10−4 10−5 10−8 10−6 1 2 3 4 5 6 7 8 9 1 (g) Errors in 2d I 2 3 d Sparse Control Problems in Measure Spaces 5 (h) Errors in 3d numerical results match a priori rates O h2− 2 Boris Vexler 4 June 2014 30 Faculty of Mathematics Technische Universität München Numerical example: improved regularity (I) I take a bounded ud as below (in 2d, α = 0.001). (i) Desired state ud I (j) Optimal state ūh (k) Optimal control q̄h the optimal state ū is bounded, optimal control q̄ is a line measure Boris Vexler Sparse Control Problems in Measure Spaces June 2014 31 Faculty of Mathematics Technische Universität München Numerical example: improved regularity (II) 100 |J(q̄, ū) − J(q̄h , ūh )| kū − ūh kL2 (Ω) 2 ε 2 kq̄h kL2 10−2 |J(q̄, ū) − J(q̄h , ūh )| kū − ūh kL2 (Ω) 2 ε 2 kq̄h kL2 10−1 10−2 h 10 10−4 h −3 10−4 −6 10 10−5 10−8 10−6 10−10 10−7 1 2 3 4 5 6 7 8 9 1 2 (l) Errors in 2d I 3 4 5 6 (m) Errors in 3d same order of convergence O(h) in 2d and 3d Boris Vexler Sparse Control Problems in Measure Spaces June 2014 32 Faculty of Mathematics Technische Universität München Outline 1. Motivation 2. Functional analytic setting 3. Optimality system and regularity 4. Discretization and error estimates 5. Parabolic case: Directional Sparsity Boris Vexler Sparse Control Problems in Measure Spaces June 2014 33 Faculty of Mathematics Technische Universität München Functional analytic setting: Choice of the control space Optimal control problem Minimize J(q, u) = 1 ku − ud k2L2 (I×Ω) + αkqkX , 2 ∂t u − ∆u = q in I × Ω, Boris Vexler q ∈ X, u ∈ V u(0) = u0 in Ω, b.c. Sparse Control Problems in Measure Spaces June 2014 34 Faculty of Mathematics Technische Universität München Functional analytic setting: Choice of the control space Optimal control problem Minimize J(q, u) = 1 ku − ud k2L2 (I×Ω) + αkqkX , 2 ∂t u − ∆u = q in I × Ω, I q ∈ X, u ∈ V u(0) = u0 in Ω, b.c. X = M(I × Ω) → sparsity in space and time, u 6∈ L2 (I × Ω)! Boris Vexler Sparse Control Problems in Measure Spaces June 2014 34 Faculty of Mathematics Technische Universität München Functional analytic setting: Choice of the control space Optimal control problem Minimize J(q, u) = 1 ku − ud k2L2 (I×Ω) + αkqkX , 2 ∂t u − ∆u = q in I × Ω, q ∈ X, u ∈ V u(0) = u0 in Ω, b.c. I X = M(I × Ω) → sparsity in space and time, u 6∈ L2 (I × Ω)! I X = L2 (I; M(Ω)) → sparsity in space, e.g. moving sources [CCK13] Boris Vexler Sparse Control Problems in Measure Spaces June 2014 34 Faculty of Mathematics Technische Universität München Functional analytic setting: Choice of the control space Optimal control problem Minimize J(q, u) = 1 ku − ud k2L2 (I×Ω) + αkqkX , 2 ∂t u − ∆u = q in I × Ω, q ∈ X, u ∈ V u(0) = u0 in Ω, b.c. I X = M(I × Ω) → sparsity in space and time, u 6∈ L2 (I × Ω)! I X = L2 (I; M(Ω)) → sparsity in space, e.g. moving sources [CCK13] I X = M(Ω; L2 (I)) → (directional) sparsity in space [PKV14] → sparsity pattern is constant in time! Boris Vexler Sparse Control Problems in Measure Spaces June 2014 34 Faculty of Mathematics Technische Universität München Control space I q ∈ X = M(Ωc ; L2 (I)) is a countably additive vector measure of bounded total variation; q : B(Ωc ) → L2 (I) Boris Vexler Sparse Control Problems in Measure Spaces June 2014 35 Faculty of Mathematics Technische Universität München Control space I I q ∈ X = M(Ωc ; L2 (I)) is a countably additive vector measure of bounded total variation; q : B(Ωc ) → L2 (I) total variation of q, |q| ∈ M(Ωc ) (∞ ) X |q|(B) = sup kq(Bn )kL2 (I) : {Bn } ⊂ B(Ωc ) disj. partition of B n=1 Boris Vexler Sparse Control Problems in Measure Spaces June 2014 35 Faculty of Mathematics Technische Universität München Control space I I q ∈ X = M(Ωc ; L2 (I)) is a countably additive vector measure of bounded total variation; q : B(Ωc ) → L2 (I) total variation of q, |q| ∈ M(Ωc ) (∞ ) X |q|(B) = sup kq(Bn )kL2 (I) : {Bn } ⊂ B(Ωc ) disj. partition of B n=1 I norm on X kqkM(Ωc ;L2 (I)) = k|q|kM(Ωc ) = |q|(Ωc ) Boris Vexler Sparse Control Problems in Measure Spaces June 2014 35 Faculty of Mathematics Technische Universität München Control space I I q ∈ X = M(Ωc ; L2 (I)) is a countably additive vector measure of bounded total variation; q : B(Ωc ) → L2 (I) total variation of q, |q| ∈ M(Ωc ) (∞ ) X |q|(B) = sup kq(Bn )kL2 (I) : {Bn } ⊂ B(Ωc ) disj. partition of B n=1 I norm on X kqkM(Ωc ;L2 (I)) = k|q|kM(Ωc ) = |q|(Ωc ) I Radon-Nikodym derivative / polar decomposition dq = q0 d|q|, i.e. Z Z ϕ(x) dq = ϕ(x)q0 (t, x) d|q| ∈ L2 (I), ∀ϕ ∈ C0 (Ωc ). Ωc Ωc with q0 ∈ L∞ (Ωc , |q|, L2 (I)), and kq0 (x)kL2 (I) = 1 for |q|-a.a. x ∈ Ωc Boris Vexler Sparse Control Problems in Measure Spaces June 2014 35 Faculty of Mathematics Technische Universität München Control space I Identify X = M(Ωc ; L2 (I)) with the dual of C(Ωc , L2 (I)) (Singer's representation theorem) Z hq, vi = (q0 (x), v(x))L2 (I) d|q|(x), q ∈ X, v ∈ C(Ωc , L2 (I)) Ωc Boris Vexler Sparse Control Problems in Measure Spaces June 2014 36 Faculty of Mathematics Technische Universität München Control space I Identify X = M(Ωc ; L2 (I)) with the dual of C(Ωc , L2 (I)) (Singer's representation theorem) Z hq, vi = (q0 (x), v(x))L2 (I) d|q|(x), q ∈ X, v ∈ C(Ωc , L2 (I)) Ωc I dense embeddings L2 (I; C(Ωc )) ,→ C(Ωc , L2 (I)) and for the dual spaces M(Ωc ; L2 (I)) ,→ L2 (I; M(Ω)) Boris Vexler Sparse Control Problems in Measure Spaces June 2014 36 Faculty of Mathematics Technische Universität München Control space I Identify X = M(Ωc ; L2 (I)) with the dual of C(Ωc , L2 (I)) (Singer's representation theorem) Z hq, vi = (q0 (x), v(x))L2 (I) d|q|(x), q ∈ X, v ∈ C(Ωc , L2 (I)) Ωc I dense embeddings L2 (I; C(Ωc )) ,→ C(Ωc , L2 (I)) and for the dual spaces M(Ωc ; L2 (I)) ,→ L2 (I; M(Ω)) I ``moving Dirac'' example: I = Ω = (0, 1), q(t) = δ{x=t} q ∈ L2 (I; M(Ω)) but q 6∈ M(Ωc ; L2 (I)) Boris Vexler Sparse Control Problems in Measure Spaces June 2014 36 Faculty of Mathematics Technische Universität München State equation State equation ∂t u − ∆u = q u=0 u(0) = 0 Boris Vexler in I × Ω, on I × ∂Ω, in Ω. Sparse Control Problems in Measure Spaces June 2014 37 Faculty of Mathematics Technische Universität München State equation State equation ∂t u − ∆u = q u=0 u(0) = 0 in I × Ω, on I × ∂Ω, in Ω. Weak formulation for u ∈ L2 (I × Ω) (u, −∂t φ − ∆φ) = hq, φi Boris Vexler ∀φ ∈ L2 (I; H2 (Ω)) ∩ H1 (I; L2 (Ω)) Sparse Control Problems in Measure Spaces June 2014 37 Faculty of Mathematics Technische Universität München State equation State equation ∂t u − ∆u = q u=0 u(0) = 0 in I × Ω, on I × ∂Ω, in Ω. Weak formulation for u ∈ L2 (I × Ω) (u, −∂t φ − ∆φ) = hq, φi I ∀φ ∈ L2 (I; H2 (Ω)) ∩ H1 (I; L2 (Ω)) For q ∈ M(Ωc ; L2 (I)) there holds u ∈ L2 (I; W01,s (Ω)) ∩ H1 (I; W −1,s (Ω)), Boris Vexler Sparse Control Problems in Measure Spaces s< d d−1 June 2014 37 Faculty of Mathematics Technische Universität München State equation State equation ∂t u − ∆u = q u=0 u(0) = 0 in I × Ω, on I × ∂Ω, in Ω. Weak formulation for u ∈ L2 (I × Ω) (u, −∂t φ − ∆φ) = hq, φi I ∀φ ∈ L2 (I; H2 (Ω)) ∩ H1 (I; L2 (Ω)) For q ∈ M(Ωc ; L2 (I)) there holds u ∈ L2 (I; W01,s (Ω)) ∩ H1 (I; W −1,s (Ω)), I s< d d−1 u ∈ L∞ (Ī; Ls (Ω)), u ∈ C(Ī; W −ε,s (Ω)) Boris Vexler Sparse Control Problems in Measure Spaces June 2014 37 Faculty of Mathematics Technische Universität München Optimal control problem Cost functional Minimize J(q, u) = 1 ku − ud k2L2 (I×Ωo ) + αkqkM(Ωc ;L2 (I)) 2 State equation ∂t u − ∆u = q u=0 u(0) = 0 Boris Vexler in I × Ω, on I × ∂Ω, in Ω. Sparse Control Problems in Measure Spaces June 2014 38 Faculty of Mathematics Technische Universität München Optimal control problem Cost functional Minimize J(q, u) = 1 ku − ud k2L2 (I×Ωo ) + αkqkM(Ωc ;L2 (I)) 2 State equation ∂t u − ∆u = q u=0 u(0) = 0 I in I × Ω, on I × ∂Ω, in Ω. Existence by (sequentional) Banach-Alaoglu theorem Boris Vexler Sparse Control Problems in Measure Spaces June 2014 38 Faculty of Mathematics Technische Universität München Optimal control problem Cost functional Minimize J(q, u) = 1 ku − ud k2L2 (I×Ωo ) + αkqkM(Ωc ;L2 (I)) 2 State equation ∂t u − ∆u = q u=0 u(0) = 0 in I × Ω, on I × ∂Ω, in Ω. I Existence by (sequentional) Banach-Alaoglu theorem I Solution operator S : M(Ωc ; L2 (I)) → L2 (I × Ω) is compact Boris Vexler Sparse Control Problems in Measure Spaces June 2014 38 Faculty of Mathematics Technische Universität München Optimal control problem Cost functional Minimize J(q, u) = 1 ku − ud k2L2 (I×Ωo ) + αkqkM(Ωc ;L2 (I)) 2 State equation ∂t u − ∆u = q u=0 u(0) = 0 in I × Ω, on I × ∂Ω, in Ω. I Existence by (sequentional) Banach-Alaoglu theorem I Solution operator S : M(Ωc ; L2 (I)) → L2 (I × Ω) is compact I Uniqueness for Ωo = Ωc . For Ωo ∩ Ωc = ∅ in general no uniqueness. Boris Vexler Sparse Control Problems in Measure Spaces June 2014 38 Faculty of Mathematics Technische Universität München Optimality system State equation ∂t u − ∆u = q u=0 u(0) = 0 in I × Ω, on I × ∂Ω, in Ω. Adjoint equation −∂t z − ∆z = (ū − ud )χΩo z=0 z(T) = 0 in I × Ω, on I × ∂Ω, in Ω. Optimality condition −hq − q̄, z̄ χΩc i + αkq̄kM(Ω;L2 (I)) ≤ αkqkM(Ω;L2 (I)) Boris Vexler Sparse Control Problems in Measure Spaces ∀q ∈ M(Ω; L2 (I)) June 2014 39 Faculty of Mathematics Technische Universität München Structure of the optimal control Theorem (a) kz̄(·, x)kL2 (I) ≤ α (b) d q̄ = ∀x ∈ Ωc q̄0 (t, x) d|q̄| and supp |q̄| ⊂ { x ∈ Ωc | kz̄(·, x)kL2 (I) = α } (c) q̄0 = − α1 χΩc z̄ almost everywhere in L∞ (Ωc , |q|, L2 (I)) Boris Vexler Sparse Control Problems in Measure Spaces June 2014 40 Faculty of Mathematics Technische Universität München Structure of the optimal control Theorem (a) kz̄(·, x)kL2 (I) ≤ α (b) d q̄ = ∀x ∈ Ωc q̄0 (t, x) d|q̄| and supp |q̄| ⊂ { x ∈ Ωc | kz̄(·, x)kL2 (I) = α } (c) q̄0 = − α1 χΩc z̄ almost everywhere in L∞ (Ωc , |q|, L2 (I)) I q̄ ∈ M(Ω; L2 (I)) ⊂ L2 (I; M(Ω)). We have q̄(t) ∈ M(Ω) and supp |q̄(t)| ⊂ supp |q̄| ⊂ { x ∈ Ωc | kz̄(·, x)kL2 (I) = α } , almost all t ∈ I I sparsity pattern constant in time Boris Vexler Sparse Control Problems in Measure Spaces June 2014 40 Faculty of Mathematics Technische Universität München Structure of the optimal control Theorem (a) kz̄(·, x)kL2 (I) ≤ α (b) d q̄ = ∀x ∈ Ωc q̄0 (t, x) d|q̄| and supp |q̄| ⊂ { x ∈ Ωc | kz̄(·, x)kL2 (I) = α } (c) q̄0 = − α1 χΩc z̄ almost everywhere in L∞ (Ωc , |q|, L2 (I)) I q̄ ∈ M(Ω; L2 (I)) ⊂ L2 (I; M(Ω)). We have q̄(t) ∈ M(Ω) and supp |q̄(t)| ⊂ supp |q̄| ⊂ { x ∈ Ωc | kz̄(·, x)kL2 (I) = α } , almost all t ∈ I I I sparsity pattern constant in time additional regularity of q̄ by (c) q̄ ∈ C(Ī; M(Ω)) Boris Vexler Sparse Control Problems in Measure Spaces June 2014 40 Faculty of Mathematics Technische Universität München Finite element discretization I Discretization concept I I state variable: dG(0) in time (implicit Euler), P1 in space control variable: dG(0) in time, Mh in space Boris Vexler Sparse Control Problems in Measure Spaces June 2014 41 Faculty of Mathematics Technische Universität München Finite element discretization I Discretization concept I I state variable: dG(0) in time (implicit Euler), P1 in space control variable: dG(0) in time, Mh in space Theorem (K. Pieper and B.V.) Let d = 2, let (q̄, ū) be the optimal solution and (q̄h , ūh ) ∈ X0 (Mh ) × Vkh the discrete optimal solution. There holds 1 kū − ūh kL2 (I×Ω) ≤ c k 2 + h |ln h|2 Boris Vexler Sparse Control Problems in Measure Spaces June 2014 41 Faculty of Mathematics Technische Universität München Finite element discretization I Discretization concept I I state variable: dG(0) in time (implicit Euler), P1 in space control variable: dG(0) in time, Mh in space Theorem (K. Pieper and B.V.) Let d = 2, let (q̄, ū) be the optimal solution and (q̄h , ūh ) ∈ X0 (Mh ) × Vkh the discrete optimal solution. There holds 1 kū − ūh kL2 (I×Ω) ≤ c k 2 + h |ln h|2 I I based on an estimate w.r.t L∞ (Ω; L2 (I)) for the adjoint state D. Leykekhman and B. Vexler Optimal a Priori Error Estimates of Parabolic Optimal Control Problems with Pointwise Control SIAM J. Num. Analysis 51, pp. 2797-2821, 2013 Boris Vexler Sparse Control Problems in Measure Spaces June 2014 41 Faculty of Mathematics Technische Universität München Finite element discretization I Discretization concept I I state variable: dG(0) in time (implicit Euler), P1 in space control variable: dG(0) in time, Mh in space Theorem (K. Pieper and B.V.) Let d = 2, let (q̄, ū) be the optimal solution and (q̄h , ūh ) ∈ X0 (Mh ) × Vkh the discrete optimal solution. There holds 1 kū − ūh kL2 (I×Ω) ≤ c k 2 + h |ln h|2 I I based on an estimate w.r.t L∞ (Ω; L2 (I)) for the adjoint state I k is independent on h D. Leykekhman and B. Vexler Optimal a Priori Error Estimates of Parabolic Optimal Control Problems with Pointwise Control SIAM J. Num. Analysis 51, pp. 2797-2821, 2013 Boris Vexler Sparse Control Problems in Measure Spaces June 2014 41 Faculty of Mathematics Technische Universität München Regularization, Optimization, Adaptivity I Regularization jγ (q) = j(q) + Boris Vexler γ kqk2L2 2 Sparse Control Problems in Measure Spaces June 2014 42 Faculty of Mathematics Technische Universität München Regularization, Optimization, Adaptivity I Regularization jγ (q) = j(q) + I γ kqk2L2 2 Error analysis kū − ūγ k ≤ cγ β Boris Vexler Sparse Control Problems in Measure Spaces June 2014 42 Faculty of Mathematics Technische Universität München Regularization, Optimization, Adaptivity I Regularization jγ (q) = j(q) + I γ kqk2L2 2 Error analysis kū − ūγ k ≤ cγ β I Semi-smooth Newton method for xed γ > 0 Boris Vexler Sparse Control Problems in Measure Spaces June 2014 42 Faculty of Mathematics Technische Universität München Regularization, Optimization, Adaptivity I Regularization jγ (q) = j(q) + I γ kqk2L2 2 Error analysis kū − ūγ k ≤ cγ β I Semi-smooth Newton method for xed γ > 0 I Adaptivity and path following γ → 0 I a posteriori error estimate for the regularization error I a posteriori error estimate for the discretization error I adaptive algorithm for path following and mesh re nement Boris Vexler Sparse Control Problems in Measure Spaces June 2014 42 Faculty of Mathematics Technische Universität München Extensions to the wave equation by young scientists Boris Vexler Sparse Control Problems in Measure Spaces June 2014 43