Pointwise convergence of the feasibility violation for Moreau-Yosida - PowerPoint PPT Presentation

Pointwise convergence of the feasibility violation for Moreau-Yosida regularized optimal control problems. Winnifried Wollner Fachbereich Mathematik Optimierung und Approximation, Universität Hamburg Workshop on Control and Optimization of PDEs Graz, October 10-14, 2011

State-Constrained optimal control W. Wollner Outline of the talk Pointwise Convergence 1 Introduction & Previous Results Error Estimates Examples Application to Gradient Constraints on the State 2 Motivation Existence & Optimality Conditions A Priori Error Estimation

State-Constrained optimal control W. Wollner Content Pointwise Convergence 1 Introduction & Previous Results Error Estimates Examples Application to Gradient Constraints on the State 2 Motivation Existence & Optimality Conditions A Priori Error Estimation

State-Constrained optimal control W. Wollner Model Problem Consider Ω ⊂ R 2 : Minimize Q × V J ( q , u ) := 1 2 � u − u d � 2 + 1 2 � q � 2 s.t. − ∆ u = q in Ω ( P ) u = 0 on ∂ Ω u ≥ u c in Ω and its regularization: Minimize Q × V J ( q , u ) + γ 2 � ( u c − u ) + � 2 ( P γ ) s.t. − ∆ u = q in Ω u = 0 on ∂ Ω (Assume throughout the existence of a Slater point)

State-Constrained optimal control W. Wollner Known Results Various contributions: Ito, Kunisch, Hintermüller, Hinze, M. Ulbrich, . . . Assuming that any solution u ∈ C (Ω) (in fact u ∈ C 0 ,β (Ω) ) gives: ( P γ ) → ( P ) as γ → ∞ Sometimes even rates for the convergence: Depends on the feasibility violation to vanish in appropriate norms. (Hintermüller, Hinze): � q − q γ � 2 ≤ C ( γ − 1 + � ( u c − u γ ) + � ∞ ) ≤ c γ − β 2 + 2 β (M. Ulbrich): � q − q γ � 2 ≤ c γ 4 13 + ǫ Looks independent of β but needs H 2 regularity

State-Constrained optimal control W. Wollner A Simple Estimate Lemma Let ( q , u ) be the solution to ( P ) and ( q γ , u γ ) the solution to ( P γ ) . Assume that it holds � ( u c − u γ ) + � ∞ ≤ c γ − θ for some θ > 0 then 0 ≤ J ( q , u ) − J ( q γ , u γ ) ≤ c γ − θ . and � q − q γ � 2 ≤ c γ − θ . hold. Lemma For the solution ( q γ , u γ ) of ( P γ ) it holds � ( u c − u γ ) + � 2 ≤ c γ − 1 � ( u c − u γ ) + � ∞ ≤ c γ − 1 − θ

State-Constrained optimal control W. Wollner A Bootstrapping Argument Lemma Let ( q γ , u γ ) be a solution of ( P γ ) . Assume that u c − u γ ∈ C 0 ,β (Ω) and that for some α > 0 it holds � ( u c − u γ ) + � 2 ≤ c γ − α then we have � ( u c − u γ ) + � ∞ ≤ c γ − α β 2 + 2 β . Theorem Under the assumptions above it holds � ( u c − u γ ) + � 2 ≤ c γ − 1 − β 2 + β then we have β � ( u c − u γ ) + � ∞ ≤ c γ − 2 + β .

State-Constrained optimal control W. Wollner Comments Results can be improved depending on the shape of the active set. Maximum attained on a line: � ( u c − u γ ) + � ∞ ≤ c γ − β 1 + β . Maximum attained on a volume: � ( u c − u γ ) + � ∞ ≤ c γ − 1 .

State-Constrained optimal control W. Wollner Example with a point measure (Cherednichenko, Krumbiegel, Rösch, 2008) 1 L-2 Norm 0.1 Point Error γ − 1 − 1 / 3 0.01 γ − 1 / 3 0.001 Feasibility violation 0.0001 1e-05 1e-06 1e-07 1e-08 1e-09 1e-10 1 10 100 1000 10000 100000 1e+06 Value of γ Figure: Convergence of � ( u c − u γ ) + � ∞

State-Constrained optimal control W. Wollner Example with a line measure (Benedix, Vexler, 2009) 100 L-2 Norm 10 Point Error γ − 1 − 3 / 4 1 γ − 3 / 4 Feasibility violation 0.1 0.01 0.001 0.0001 1e-05 1e-06 1 10 100 1000 10000 Value of γ Figure: Convergence of � ( u c − u γ ) + � ∞

State-Constrained optimal control W. Wollner Content Pointwise Convergence 1 Introduction & Previous Results Error Estimates Examples Application to Gradient Constraints on the State 2 Motivation Existence & Optimality Conditions A Priori Error Estimation

State-Constrained optimal control W. Wollner Model Problem Consider: Minimize Q × V 1 2 � u − u d � 2 + α r � q � r L r s.t. − ∆ u = q in Ω ( P ) u = 0 on ∂ Ω |∇ u | 2 ≤ c in Ω Discretize with FEM A priori error analysis: Deckelnick, Günther, Hinze; Ortner, Wollner � q − q h � r = O ( h 1 − 2 / t ) Need a ‘smooth’ domain to have u ∈ C 1 (Ω) . What if Ω ⊂ R 2 is not smooth but is a (nonconvex) polygon?

State-Constrained optimal control W. Wollner Model Problem Consider: Minimize Q × V 1 2 � u − u d � 2 + α r � q � r L r s.t. − ∆ u = q in Ω ( P ) u = 0 on ∂ Ω |∇ u | 2 ≤ c in Ω Discretize with FEM A priori error analysis: Deckelnick, Günther, Hinze; Ortner, Wollner � q − q h � r = O ( h 1 − 2 / t ) Need a ‘smooth’ domain to have u ∈ C 1 (Ω) . What if Ω ⊂ R 2 is not smooth but is a (nonconvex) polygon?

State-Constrained optimal control W. Wollner Existence The solution u of the state equation can be rewritten (one corner, a.a. t ): u = u r + u s u r ∈ W 2 , t (Ω) ∩ H 1 0 (Ω) ⊂ C 1 (Ω) ( t > 2) u s = � j < 2 ω c α j r j π/ω ϕ ( θ ) π t ′ j = 1 j π ω c � = 1 u ∈ W 1 , ∞ (Ω) implies α j = 0 And − ∆ : W := W 2 , t (Ω) ∩ H 1 0 (Ω) �→ I = ∆ W has a continuous inverse. → Solution to ( P ) by standard arguments on the control space Q = I ∩ L r (Ω) .

State-Constrained optimal control W. Wollner Necessary Conditions For ( q , u ) ∈ I × W there exist µ ∈ C (Ω) ∗ and a function z ∈ L t ′ (Ω) s.t. ∀ ϕ ∈ H 1 ( ∇ u , ∇ ϕ ) = ( q , ϕ ) 0 (Ω) , �− ∆ ϕ, z � = ( u − u d , ϕ ) + � µ, ∇ u ∇ ϕ � ∀ ϕ ∈ W , ( q | q | r − 2 , δ q ) = −� δ q , z � ∀ δ q ∈ Q ∩ I , � µ, ϕ � ≤ 0 ∀ ϕ ∈ C (Ω) , ϕ ≤ 0 , 0 = � µ, |∇ u | 2 − c � . Note that z is determined modulo I ⊥ only. 1 − 2 / t r − 1 , r (Ω) → q ∈ W

State-Constrained optimal control W. Wollner Discretization & Simple Error Estimates Minimize I × V h J ( q h , u h ) ( P ⊥ h ) s.t. ( ∇ u h , ∇ ϕ ) = ( q , ϕ ) ∀ ϕ ∈ V h |∇ u h | 2 ≤ c in Ω We follow the ideas of Ortner, Wollner Numer. Math. (2011) Theorem Let ( q , u ) be solutions to ( P ) and ( q h , u h ) solutions to ( P ⊥ h ) then there exists t > 2 and a constant C such that for any ǫ > 0 : h ) | ≤ Ch 1 − 2 / t − ǫ =: Ch β | J ( q , u ) − J ( q ⊥ h , u ⊥ and r ≤ Ch β � q − q h � r

State-Constrained optimal control W. Wollner Regularization Minimize Q × V h J ( q h , u h ) ( P h ) s.t. ( ∇ u h , ∇ ϕ ) = ( q , ϕ ) ∀ ϕ ∈ V h |∇ u h | 2 ≤ c in Ω Problem: Continuous solution u h to q h is not in W 1 , ∞ . Minimize Q × V J ( q , u ) = J ( q , u ) + γ 2 � ( |∇ u | 2 − c ) + � 2 ( P γ ) s.t. ( ∇ u , ∇ ϕ ) = ( q , ϕ ) ∀ ϕ ∈ V |∇ u | 2 ≤ c in Ω Observation: � ( |∇ u h | 2 − c ) + � 2 ≤ ch 2 π/ω Allows to estimate the distance between ( P h ) and ( P γ ) .

State-Constrained optimal control W. Wollner Error Estimates Theorem Assume that γ | 2 − c ) + � ∞ ≤ c γ − θ � ( |∇ u r holds then | J ( q γ , u γ ) − J ( q , u ) | ≤ c γ − θ Lemma It holds − 1 − π/ω | α γ | ≤ c γ 1 + π/ω γ | 2 − c ) + � ∞ ≤ c γ β 1 − π/ω − � ( |∇ u r 1 + β 1 + π/ω Note that the rates can be improved depending on the type of the active set.

State-Constrained optimal control W. Wollner Error Estimates - cont. Theorem It holds for the solution ( q h , u h ) to ( P h ) and ( q , u ) to ( P ) that β 2 π/ω ( 1 − π/ω ) | J ( q , u ) − J ( q , u ) | ≤ ch 1 + π/ω + 2 β

State-Constrained optimal control W. Wollner Example Optimal State State Unconstraint

State-Constrained optimal control W. Wollner Example Optimal Control on Mesh 5 Optimal Control on Mesh 6

State-Constrained optimal control W. Wollner Error in the Cost Functional Global refinement Local refinement O ( N − 0 . 25 ) 0.1 Relativ Functional error 0.01 100 1000 10000 100000 Nodes Figure: Error in the cost functional

State-Constrained optimal control W. Wollner Conclusions and Outlook New (improved) error estimates for Moreau-Yosida regularized state constraints. Shown existence and necessary optimality conditions on non smooth domains. Derived error estimates for the discretization error. Thank you for your attention!

State-Constrained optimal control W. Wollner Conclusions and Outlook New (improved) error estimates for Moreau-Yosida regularized state constraints. Shown existence and necessary optimality conditions on non smooth domains. Derived error estimates for the discretization error. Thank you for your attention!