Numerical Solution of Elliptic Optimal Control Problems

Transcription

1 Department of Mathematics, University of Houston Numerical Solution of Elliptic Optimal Control Problems Ronald H.W. Hoppe 1,2 1 Department of Mathematics, University of Houston 2 Institute of Mathematics, University of Augsburg Summer School Optimal Control of PDEs Cortona, July 12-17, 2010

2 Department of Mathematics, University of Houston Recent Books on Optimal Control of PDEs R. Glowinski, J.L. Lions, and J. He; Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach. Cambridge University Press, Cambridge, M. Hinze, R. Pinnau, and M. Ulbrich; Optimization with PDE Constraints. Springer, Berlin-Heidelberg-New York, F. Tröltzsch; Optimal Control of Partial Differential Equations. Theory, Methods, and Applications. American Mathematical Society, Providence, 2010.

3 Numerical Solution of Elliptic Optimal Control Problems Unconstrained Elliptic Optimal Control Problems Control Constrained Elliptic Optimal Control Problems State Constrained Elliptic Optimal Control Problems

4 Department of Mathematics, University of Houston Elliptic Optimal Control Problems Unconstrained Case

5 Elliptic Optimal Control Problems: Unconstrained Case Consider the unconstrained distributed elliptic optimal control problem inf J(y,u) := 1 y y d 2 dx + α u 2 dx, y,u 2 2 where u L 2 (Ω). Ω (EUC) 1 y = u in Ω, (EUC) 2 y = 0 on Γ, Ω

6 Existence and Uniqueness of a Solution We introduce the control-to-state map G : L 2 (Ω) H 1 0(Ω), y = Gu solves y = u. Substituting y = Gu allows to consider the reduced control problem inf u J red(u) := 1 2 Gu yd 2 L 2 (Ω) + α 2 u 2 L 2 (Ω). Theorem (Existence and uniqueness) The unconstrained elliptic control problem admits a unique solution y H 1 0(Ω). Proof. Direct method of the calculus of variations (minimizing sequence).

7 Optimality Conditions for the Distributed Control Problem There exists an adjoint state p H 1 0(Ω) such that the triple (y,p,u) satisfies a(y,v) = (u,v) L 2 (Ω), v H 1 0(Ω), a(p,v) = (y y d,v) L 2 (Ω), v H 1 0(Ω), p αu = 0, where a(, ) stands for the bilinear form a(w,z) = Ω w z dx, w,z H 1 0(Ω).

8 Optimality Conditions for the FE Discretized Control Problem There exists an adjoint state p h V h such that the triple (y h,p h,u h ) satisfies a(y h,v h ) = (u h,v h ) L 2 (Ω), v h V h, a(p h,v h ) = (y h y d,v h ) L 2 (Ω), v h V h, p h αu h = 0.

9 Finite Element Approximation of the Distributed Control Problem Let T h (Ω) be a shape regular, simplicial triangulation of Ω and let V h := { v h C(Ω) v h T P 1 (T), T T h (Ω), v h Ω = 0 } be the FE space of continuous, piecewise linear finite elements. Consider the following FE Approximation of the distributed control problem Minimize J(y h,u h ) := 1 2 y h y d 2 L 2 (Ω) + α 2 u h 2 L 2 (Ω), over (y h,u h ) V h V h, subject to a(y h,v h ) = (u h,v h ) L 2 (Ω), v h V h.

10 Algebraic Formulation of the Discrete Optimality System Let V l := span{ϕ l (1),,ϕ(N l ) l }, where ϕ (i) l,1 i N l, are the nodal basis functions. We refer to M l lr N l N l and A l lr N l N l as the associated mass matrix and stiffness matrix: (M l ) ij := (ϕ (i) l,ϕ(j) l ) L 2 (Ω), (A l ) ij := a(ϕ (i) l, ϕ(j) l ), 1 i,j N l. The solution of the discrete optimality system requires the computation of (y l,p l ) lr N l lr N l as the solution of A l α 1 M l M l A l y l p l = 0 M l y d l.

11 Left and Right Transforming Iterations For given K lr N N,d lr N, we want to solve the linear algebraic system ( ) Kx = d. Let K L,K R lr N N be regular. Then, ( ) can be equivalently wriiten as ( ) K L KK R K 1 R x = K L d. Assuming K to be a suitable preconditioner for K L KK R, we consider the transforming iteration K 1 R x (ν+1) = K 1 R x (ν) + K 1 (K L d K L Kx (ν) ). Backtransformation yields so that ˆK := K 1 L x (ν+1) = x (ν) + K R K 1 K }{{ L (d Kx (ν) ), } = (K 1 L KK 1 R ) 1 1 KK R is an appropriate preconditioner for the original system.

12 Department of Mathematics, University of Houston Left and Right Transforming Iterations II We apply the idea of left/right transformations to K = A l α 1 M l M l A l, d = 0 M l y d l We choose the left and right transformations according to K L = α 1/2 I 0 0 I, K R = α 1/2 I 0 0 I.. This leads to the transformed matrix K L KK R = A l α 1/2 M l α 1/2 M l A l.

13 Left and Right Transforming Iterations III Denoting by S l the Schur complement S l = A l + α 1 M l A 1 l M l, we have A l α 1/2 M l A α 1/2 = l α 1/2 M l M l A l α 1/2 M l S l + α 1 M l A 1 l M. l Choosing  l as a preconditioner for A l and Ŝ l := τ 1 l diag(a l + α 1 M l  1 l M l ), we obtain the symmetric Uzawa preconditioner K l =  l α 1/2 M l α 1/2 M l Ŝ l + α 1 M l  1 l M l Backtransformation results in the following preconditioner ˆK l = K 1 L K l K 1 R =.  l α 1 M l M l Ŝ l α 1 M l  l 1 M l.

14 Multigrid Solvers for Elliptic Optimal Control Problems A. Borzi, K. Kunisch, and D. Y. Kwak; Accuracy and convergence properties of the finite difference multigrid solution of an optimal control optimality system. SIAM J. Control Optimization 41, , A. Borzi and V. Schulz; Multigrid methods for PDE optimization. SIAM Rev. 51, , J. Schöberl, R. Simon, and W. Zulehner; A robust multigrid method for elliptic optimal control problems. Preprint, Inst. of Comput. Math., University of Linz, 2010.

15 Department of Mathematics, University of Houston Step 1: Given iterates y (ν) l,p(ν) l Preconditioned Richardson Iteration I Res (1) l Res (2) l lr N l, compute the residuals = 0 M l y d l Step 2: Solve the linear system  l α 1 M l M l Ŝ l α 1 M l  1 l M l Step 3: Compute the new iterates y (ν+1) l p (ν+1) l = y (ν) l p (ν) l K l y (ν) l p (ν) l + y (ν) l p (ν) l = y (ν) l p (ν) l.. Res (1) l Res (2) l.

16 Department of Mathematics, University of Houston Consider the linear system ( ) Preconditioned Richardson Iteration II  l α 1 M l M l Ŝ l α 1 M l  1 l M l y (ν) l p (ν) l = Res (1) l Res (2) l. Elimination of y l (ν) results in the Schur complement system Ŝ l p l (ν) = Res l (2) Ml 1 l Res(1) l. Hence, the solution of ( ) can be reduced to the successive solution of the three linear systems  l y l (ν) = Res l (1), Ŝ l p (ν) l = Res l (2) M (ν) l y l,  l y (ν) l = Res (1) l + α 1 M l p (ν) l.

17 Preconditioned Richardson Iteration III Theorem [Schöberl/Simon/Zulehner] If the preconditioners  l and Ŝ l are chosen according to then there holds  l = σ 1 l diag(a l ), Ŝ l = τ 1 l diag(s l ), σ l λ max ((diag(a l )) 1 A l ) 1, τ l λ max ((diag(s l )) 1 S l ) 1, where (y l (ν),p(ν) l ) (y l,p l ) C (1 + α 1/2 h 2 l ) η(ν) (y(0) l,p(0) l ) (y l,p l ), η(ν) 2 π(ν 1) 2 πν, if ν is even,, if ν is odd and C is a positive constant independent of h l and α.

18 Multigrid Solution of the Optimality System The previous theorem suggests to use the preconditioned iterative solver as a smoother within a multigrid solution of the optimality system. Theorem. In case of two nested grids T l 1 (Ω) and T l (Ω) let (y l (0),p(0) l ) be a level l startiterate and let (y l (1),p(1) l ) be the result of a two-grid cycle with ν > 0 smoothing steps. Then there holds (y l (1),p(1) l ) (y l,p l ) C 1 + α 1/2 h 2 l (y (0) ν l,p(0) l ) (y l,p l ), where C > 0 is a constant independent of h l and α.

19 Department of Mathematics, University of Houston Elliptic Optimal Control Problems Control Constraints

20 Elliptic Optimal Control Problems: Control Constraints Consider the control constrained distributed elliptic optimal control problem inf J(y,u) := 1 y y d 2 dx + α u 2 dx, y,u 2 2 and where u L 2 (Ω). Ω (ECC) 1 y = u in Ω, (ECC) 2 y = 0 on Γ, u K C := {v L 2 (Ω) v(x) ψ(x) f.a.a. x Ω}, Ω

21 Control Constraints: Necessary Optimality Conditions Theorem. If (y,u) H 1 0(Ω) K C is the optimal solution of the optimal control problem, then there exists (p,λ) H 1 0(Ω)) L 2 (Ω) such that p satisfies the adjoint state equation and there holds p = (y y d ) in Ω, p = 0 on Γ, p λ = αu in Ω, λ I Kc (u).

22 Department of Mathematics, University of Houston Elliptic Optimal Control Problems Primal-Dual Active Set Strategy

23 Moreau-Yosida Approximation of Multivalued Maps I Weighted Duality Mapping: Assume that V is a Banach space with dual V and let h : R + R + be a continuous and non-decreasing function such that h(0) = 0 and h(t) as t. Then the mapping J h : V 2 V J h (u) := {u V u,u = u u, u = h( u )} is called the duality mapping with weight h. Example: For V = L p (Ω),V = L q (Ω),1 < p,q <,1/p +1/q = 1, and h(t) = t p 1 we have J h (u)(x) = { u(x) p 1 sgn(u(x)), u(x) 0 0, u(x) = 0.

24 Moreau-Yosida Approximation of Multivalued Maps II Moreau-Yosida proximal map: Let f : V R be a lower semi-continuous proper convex function with subdifferential f. For c > 0, the Moreau-Yosida proximal map P f c : V 2 V is defined such that P f c (w), w V, is the set of minimizers of where j h = J h. inf f(v) + cj h( v w ), v V c Moreau-Yosida approximation: If J h is single-valued, then for c > 0 the Moreau- Yosida approximation ( f) c of f is given by ( f) c (w) := J h (c 1 w c 1 P f c (w)).

25 Department of Mathematics, University of Houston Moreau-Yosida Approximation of I KC Idea: Approximate I KC by its Moreau-Yosida approximation ( I KC ) c. Theorem. For any c > 0, we have λ ( I KC ) c, if and only if there holds ( ) λ = c u + c 1 λ Π KC (u + c 1 λ) = c max(0,u + c 1 λ ψ), and this is equivalent to u = Π KC (u + c 1 λ), where Π KC denotes the L 2 -projection onto K C.

26 Step 1 (Initialization): Primal-Dual Active Set Strategy I Choose c > 0, start-iterates y (0) h,u(0) h,λ(0) h and set n = 1. Step 2 (Specification of active/inactive sets): Compute the active/inactive sets A n and I n according to A n := {1 i N (u (n 1) h Step 3 (Termination criterion): + c 1 λ (n 1) h ) i > (ψ h ) i }, I n := {1,,N} \ A n. If n 2 and A n = A n 1, stop the algorithm. Otherwise, go to Step 4.

27 Department of Mathematics, University of Houston Primal-Dual Active Set Strategy II Step 4 (Update of the state, adjoint state, and control): Compute y (n) h,p(n) h and set (A h y (n) h ) i = as the solution of { (ψ h ) i, if i A n α 1 (M 1 h p, A h) i, if i I h p (n) h n { (u (n) h ) i := Step 5 (Update of the multiplier): Set λ (n) h (ψ h ) i, if i A n α 1 (M 1 h p(n) h ). i, if i I n := p(n) h αm hu (n) h, n := n + 1, and go to Step 2. = M h(y h y d h),

28 PDAS As A Semi-Smooth Newton Method I Newton differentiability: A function f : S X Y (X, Y Banach spaces) is called Newton differentiable at x S, if there exist a neighborhood U(x) S and a family of mappings g : U(x) L(X, Y) such that f(x + h) f(x) g(x + h)(h) Y = o( h X ) g(x) is said to be the generalized derivative of f at x. (as h X ). Example: The max-function max : L p (Ω) L q (Ω) is Newton differentiable for 1 q < p <. If f : L r (Ω) L p (Ω) is Fréchet-differentiable for some 1 < r, then the function x χ A (x) f(u(x)) is the generalized derivative of max(0,f( )). Here, χ A denotes the characteristic function of the set A where f(u( )) is nonnegative.

29 PDAS As A Semi-Smooth Newton Method II Setting z h := (y h,u h,λ h ), the discrete optimality conditions require the solution of the nonlinear system A h y h M h u h 0 F h (z h ) := αa h u h + M h y h yh d + λ h = 0. λ h + c max(0,u h + c 1 λ h ψ h ) 0 The primal-dual active set strategy (PDAS) is equivalent to the semi-smooth Newton method F h(z (n) h ) z(n) h z (n+1) h = F h (z (n) h ), = z (n) h + z(n) h, where F h (z(n) h ) is the generalized derivative of F h at z (n) h.

30 Department of Mathematics, University of Houston Elliptic Optimal Control Problems Interior Point Methods

31 Interior-Point Methods (General) Gonzaga [1992], Wright [1992], Zhang/Tapia/Dennis [1992] Conn/Gould/Toint [1996], Heinkenschloss [2001] Forsgren/Gill/Wright [2002], M. Ulbrich [2002], S. Ulbrich [2003] Interior-Point Methods (PDE Constrained Optimization) Ulbrich/Ulbrich/Heinkenschloss [1999], Ulbrich/Ulbrich [2000] H./Petrova/Schulz [2002], H./Petrova [2004], Weiser [2005] Ulbrich/Ulbrich [2007], Antil/H./Linsenmann [2007]

32 Interior Point Method I Idea: Coupling of control constraints u h ψ h by, e.g., logarithmic barrier functions with barrier parameter β > 0: N inf (y h,u h ) := J h (y h,u h ) + β ln((ψ h u h ) i ), B (β) y h,u h h which represents a parameter-dependent family of minimization subproblems. Coupling of the discrete state equation by a Lagrange multiplier: i=1 inf sup y h,u h L (β) h p h (y h,u h,p h ) := B (β) h (y h,u h ) + p h,a h y h M h u h.

33 Interior Point Method II The barrier path β x h (β) := (y h (β),u h (β),p h (β)) is the solution of the parameterdependent nonlinear system: A h p h + M h y h y d h A h y h M h u 0 h β F h (x h (β),β) = α(m h u h ) (ψ h u h ) 1 = 0. β α(m h u h ) N + 0 (ψ h u h ) N Solution by path-following predictor-corrector continuation methods in the barrier parameter.

34 Continuation Strategies in Interior-Point Methods (i) Static Strategies Fiacco/McCormick [1990], Byrd/Hribar/Nocedal [1999] Wächter/Biegler [2005] (ii) Dynamic Update Strategies El-Bakry/Tapia/Tsuchiya/Zhang [1996], Gay/Overton/Wright [1998] Vanderbei/Shanno [1999], H./Petrova/Schulz [2002] Tits/Wächter/Bakhtiari/Urban/Lawrence [2003], H./Petrova [2004] Ulbrich/Ulbrich/Vicente [2004], H./Linsenmann/Petrova [2006] Nocedal/Wächter/Waltz [2006], Armand/Benoist/Orban [2007]

35 Department of Mathematics, University of Houston The Barrier Path x( 0 ) Central Path x* "contraction tube"

36 Department of Mathematics, University of Houston Barrier Path: The Predicted Continuation Step x??? x k-1 k x( ) k-1 x( 0 ) x* may leave the Kantorovich neighborhood.

37 Department of Mathematics, University of Houston Barrier Path: The Newton Correction new x k-1 k x x( k ) x( k-1) x( 0 ) x* reduces the continuation steplength adaptively.

38 Numerical Results: Distributed Control Problem with Control Constraints I Minimize J(y,u) := 1 2 y yd 2 0,Ω + α 2 u 2 0,Ω over (y,u) H 1 0(Ω) L 2 (Ω) subject to y = u in Ω, u K := {v L 2 (Ω) v ψ a.e. in Ω} Data: Ω := (0,1) 2, { y d 200 x := 1 x 2 (x 1 0.5) 2 (1 x 2 ), 0 x (x 1 1) (x 2 (x 1 0.5) 2 (1 x 2 ), 0.5 < x 1 1, α = 0.01, ψ = 1.

39 Numerical Results: Distributed Control Problem with Control Constraints I Optimal state (left) and optimal control (right)

40 Numerical Results: Distributed Control Problem with Control Constraints I Grid after 6 (left) and 10 (right) refinement steps

41 Numerical Results: Distributed Control Problem with Control Constraints II Minimize J(y,u) := 1 2 y yd 2 0,Ω + α 2 u ud 2 0,Ω over (y,u) H 1 0(Ω) L 2 (Ω) subject to y = f + u in Ω, u K := {v L 2 (Ω) v ψ a.e. in Ω} Data: Ω := (0,1) 2, y d := 0, u d := û + α 1 (ˆσ 2 û), { (x1 0.5) ψ := 8, (x 1,x 2 ) Ω 1, (x 1 0.5) 2, α := 0.1, f := 0, otherwise { { ψ, (x û := 1,x 2 ) Ω 1 Ω 2, 2.25 (x1 0.75) 10, ˆσ := 4, (x 1,x 2 ) Ω 2, 1.01 ψ, otherwise 0, otherwise ( ) 1/2 Ω 1 := {(x 1,x 2 ) Ω (x 1 0.5) 2 + (x 2 0.5) }, Ω2 := {(x 1,x 2 ) Ω x }.,

42 Numerical Results: Distributed Control Problem with Control Constraints II Optimal state (left) and optimal control (right)

43 Numerical Results: Distributed Control Problem with Control Constraints II Grid after 6 (left) and 10 (right) refinement steps

44 Department of Mathematics, University of Houston Elliptic Optimal Control Problems State Constraints

45 Literature on State-Constrained Optimal Control Problems M. Bergounioux, K. Ito, and K. Kunisch (1999) M. Bergounioux, M. Haddou, M. Hintermüller, and K. Kunisch (2000) M. Bergounioux and K. Kunisch (2002) E. Casas (1986) E. Casas and M. Mateos (2002) J.-P. Raymond and F. Tröltzsch (2000) K. Deckelnick and M. Hinze (2006) M. Hintermüller and K. Kunisch (2007) K. Kunisch and A. Rösch (2002) C. Meyer and F. Tröltzsch (2006) C. Meyer, U. Prüfert, and F. Tröltzsch (2005) U. Prüfert, F. Tröltzsch, and M. Weiser (2004) H./M. Kieweg (2007) A. Günther, M. Hinze (2007) O. Benedix, B. Vexler (2008) M. Hintermüller/H. (2008) W. Liu, W. Gong and N. Yan (2008)

46 Elliptic Optimal Control Problems: State Constraints Consider the state constrained distributed elliptic optimal control problem inf J(y,u) := 1 y y d 2 dx + α u 2 dx, y,u 2 2 and where u L 2 (Ω). Ω (ESC) 1 y = u in Ω, (ESC) 2 y = 0 on Γ, y K S := {y H 1 0(Ω) y(x) ψ(x) f.a.a. x Ω}, Ω

47 State Constraints: Necessary Optimality Conditions Theorem. The state constrained optimal control problem admits a unique solution (y,u) with y W 1,r 0 (Ω),r > 2, and u L2 (Ω). Theorem. Let us assume that there exists u 0 L 2 (Ω) such that the associated solution y 0 W 1,r 0 (Ω) of the state equation satisfies y 0 int(k S ) (Slater condition). If (y,u) is the unique solution of the state constrained optimal control problem, there exist p W 1,s 0 (Ω),1/r + 1/s = 1, and λ M +(Ω) such that ( p, v) L 2 (Ω) = (y y d,v) L 2 (Ω) + λ,v, v W 1,r 0 (Ω), p = αu, λ,y ψ = 0.

48 Department of Mathematics, University of Houston Finite Element Approximation Let T h (Ω) be a simplicial triangulation of Ω and let V h := { v h C(Ω) v h T P 1 (T), T T h (Ω), v h Γ = 0 } be the FE space of continuous, piecewise linear functions. Let ψ h be the V h -interpoland of ψ. Consider the following FE Approximation of the state constrained control problem Minimize J h (y h,u h ) := 1 2 y h y d 2 0,Ω + α 2 u h 2 0,Ω, over (y h,u h ) V h V h, subject to ( y h, v h ) 0,Ω = (u h,v h ) 0,Ω, v h V h, y h K h,s := {v h V h v h (x) ψ h (x), x Ω}. Since the constraints are point constraints associated with the nodal points, the discrete multipliers are chosen from M h := {λ h M(Ω) λ h = a N h (Ω) κ a δ a, κ a lr}.

49 Step 1 (Initialization): Primal-Dual Active Set Strategy I Choose c > 0, start-iterates y (0) h,u(0) h,λ(0) h and set n = 1. Step 2 (Specification of active/inactive sets): Compute the active/inactive sets A n and I n according to A n := {1 i N (y (n 1) h Step 3 (Termination criterion): + c 1 λ (n 1) h ) i > (ψ h ) i }, I n := {1,,N} \ A n. If n 2 and A n = A n 1, stop the algorithm. Otherwise, go to Step 4.

50 Department of Mathematics, University of Houston Primal-Dual Active Set Strategy II Step 4 (Update of the state, adjoint state, control, and multiplier): Compute (y (n) h,u(n) h,p(n) h, λ(n) h ) as the solution of A h y (n) h A h p (n) h (M hy (n) h Set n := n + 1 and go to Step 2. M hu (n) h = 0, yd h) λ (n) h = 0, p (n) h αm hu (n) h = 0, (y (n) h ) i = (ψ h ) i for i A n, (λ (n) h ) i = 0 for i A n.

51 Lavrentiev Regularization: Mixed Control-State Constraints Introduce a regularization parameter ε > 0 and consider the mixed controlstate constrained optimal control problem Minimize J(y ε,u ε ) := 1 2 yε y d 2 L 2 (Ω) + α 2 uε 2 L 2 (Ω), subject to y ε = u ε in Ω, y ε = 0 on Γ D, εu ε + y ε K := {v L 2 (Ω) v(x) ψ(x), f.a.a. x Ω}. Theorem (Optimality conditions). The optimal solution (y ε,u ε ) V L 2 (Ω) is characterized by the existence of an adjoint state p ε V and a multiplier λ ε L 2 +(Ω) such that ( y ε, v) L 2 (Ω) = (u ε,v) L 2 (Ω), v V, ( p ε, v) L 2 (Ω) = (y ε y d,v) L 2 (Ω) + (λ ε,v) L 2 (Ω), v V, p ε α u ε + ελ ε = 0, (λ ε, εu ε + y ε ψ) L 2 (Ω) = 0.

52 Numerical Results: Distributed Control Problem with State Constraints I Minimize J(y,u) := 1 2 y yd 2 0,Ω + α 2 u ud 2 0,Ω over (y,u) H 1 0(Ω) L 2 (Ω) subject to y = u in Ω, y K := {v H 1 0(Ω) v ψ a.e. in Ω} Data: Ω := ( 2, +2) 2, y d (r) := y(r) + p(r) + σ(r), u d (r) := u(r) + α 1 p(r), ψ := 0, α := 0.1, where y(r),u(r),p(r), σ(r) is the solution of the problem: y(r) := r 4/3 + γ 1 (r), u(r) = y(r), p(r) = γ 2 (r) + r r r2, σ(r) :=, γ 1 := γ 2 := 1, r < (r 0.25) (r 0.25) 4 80(r 0.25) 3 + 1, 0.25 < r < , otherwise { 1, r < , otherwise. { 0.0, r < , otherwise,

53 Numerical Results: Distributed Control Problem with State Constraints I Optimal state (left) and optimal control (right)

54 Department of Mathematics, University of Houston Numerical Results: Distributed Control Problem with State Constraints II Minimize J(y,u) := 1 2 y yd 2 0,Ω + α 2 u ud 2 0,Ω over (y,u) H 1 (Ω) L 2 (Ω) subject to y + cy = u in Ω, y K := {v H 1 (Ω) v ψ a.e. in Ω} Data: Ω = B(0,1) := {x = (x 1,x 2 ) T x x 2 2 < 1}, y d (r) := π 1 4π r π ln(r), u d (r) := π r2 1 ln(r), ψ := 4 + r, α := 1. 2π The solution y(r),u(r),p(r), σ(r) of the problem is given by y(r) 4, u(r) 4, p(r) = 1 4π r2 1 2π ln(r), σ(r) = δ 0.

55 Numerical Results: Distributed Control Problem with State Constraints II Optimal state (left) and optimal control (right)

56 Numerical Results: Distributed Control Problem with State Constraints II Optimal adjoint state (left) and mesh after 16 adaptive loops (right)