Optimal control of AllenCahn equations with singular potentials and - PowerPoint PPT Presentation

Weierstrass Institute for Applied Analysis and Stochastics Optimal control of Allen–Cahn equations with singular potentials and dynamic boundary conditions Jürgen Sprekels (joint work with P . Colli) Mohrenstrasse 39 · 10117 Berlin · Germany · Tel. +49 30 20372 0 · www.wias-berlin.de DIMO 2013

The optimal control problem Consider the IVP with dynamic boundary condition y t − ∆ y + f ′ ( y ) = u a.e. in Q , (1) ∂ t y Γ − ∆ Γ y Γ + ∂ n y + g ′ ( y Γ ) = u Γ , y | Γ = y Γ , a.e. on Σ , (2) y ( 0 ) = y 0 a.e. in Ω , y Γ ( 0 ) = y 0 Γ a.e. on Γ . (3) Here, we have n : ∆ Γ : Laplace–Beltrami operator, outward unit normal derivative; � f , g : given nonlinearities; � u , u Γ : control functions; � y 0 ∈ H 1 ( Ω ) : initial datum s.t. y 0 | Γ = y 0 Γ . � Optimal control ... · DIMO 2013 · Page 2 (26)

The optimal control problem We introduce the Banach spaces H : = L 2 ( Ω ) , V : = H 1 ( Ω ) , H Γ : = L 2 ( Γ ) , V Γ : = H 1 ( Γ ) , H : = L 2 ( Q ) × L 2 ( Σ ) , X : = L ∞ ( Q ) × L ∞ ( Σ ) , � ( y , y Γ ) : y ∈ H 1 ( 0, T ; H ) ∩ C 0 ([ 0, T ] ; V ) ∩ L 2 ( 0, T ; H 2 ( Ω )) , Y : = � y Γ ∈ H 1 ( 0, T ; H Γ ) ∩ C 0 ([ 0, T ] ; V Γ ) ∩ L 2 ( 0, T ; H 2 ( Γ )) , y Γ = y | Γ , endowed with their respective natural norms. We also assume: (A1) There are given functions z Q ∈ L 2 ( Q ) , z Σ ∈ L 2 ( Σ ) , z T ∈ V , z Γ , T ∈ V Γ , u 2 ∈ L ∞ ( Q ) with � u 1 , � � u 1 ≤ � u 2 a.e. in Q , u 2 Γ ∈ L ∞ ( Σ ) with � u 1 Γ , � � u 1 Γ ≤ � u 2 Γ a.e. on Σ . Optimal control ... · DIMO 2013 · Page 3 (26)

The optimal control problem (CP) Minimize the (tracking-type) cost functional T T � � � � � � J (( y , y Γ ) , ( u , u Γ )) : = β 1 � 2 d x d t + β 2 | y Γ − z Σ | 2 d Γ d t � y − z Q 2 2 0 0 Ω Γ � � + β 3 | y ( · , T ) − z T | 2 d x + β 4 | y Γ ( · , T ) − z Γ , T | 2 d Γ 2 2 Ω Γ T T � � � � + β 5 | u | 2 d x d t + β 6 | u Γ | 2 d Γ d t (4) 2 2 0 Ω 0 Γ subject to the state system (1)–(3) and to the control constraint ( u , u Γ ) ∈ U ad : = { ( w , w Γ ) ∈ H : � u 1 ≤ w ≤ � u 2 a.e. in Q , u 1 Γ ≤ w Γ ≤ � � a.e. on Σ } . u 2 Γ (5) Optimal control ... · DIMO 2013 · Page 4 (26)

General assumptions f = f 1 + f 2 , g = g 1 + g 2 , where f 2 , g 2 ∈ C 3 [ 0, 1 ] , and where (A2) f 1 , g 1 ∈ C 3 ( 0, 1 ) are convex and satisfy: r ց 0 f ′ r ց 0 g ′ r ր 1 f ′ r ր 1 g ′ 1 ( r ) = lim 1 ( r ) = − ∞ , 1 ( r ) = lim 1 ( r ) = + ∞ lim lim (6) | f ′ 1 ( r ) | ≤ M 1 + M 2 | g ′ ∃ M 1 ≥ 0, M 2 > 0 : 1 ( r ) | ∀ r ∈ ( 0, 1 ) . (7) Optimal control ... · DIMO 2013 · Page 5 (26)

General assumptions f = f 1 + f 2 , g = g 1 + g 2 , where f 2 , g 2 ∈ C 3 [ 0, 1 ] , and where (A2) f 1 , g 1 ∈ C 3 ( 0, 1 ) are convex and satisfy: r ց 0 f ′ r ց 0 g ′ r ր 1 f ′ r ր 1 g ′ 1 ( r ) = lim 1 ( r ) = − ∞ , 1 ( r ) = lim 1 ( r ) = + ∞ lim lim (6) | f ′ 1 ( r ) | ≤ M 1 + M 2 | g ′ ∃ M 1 ≥ 0, M 2 > 0 : 1 ( r ) | ∀ r ∈ ( 0, 1 ) . (7) y 0 Γ = y 0 | Γ , f 1 ( y 0 ) ∈ L 1 ( Ω ) , g 1 ( y 0 Γ ) ∈ L 1 ( Γ ) , and y 0 ∈ V , (A3) 0 < y 0 < 1 a.e. in Ω , 0 < y 0 Γ < 1 a.e. on Γ . (8) Optimal control ... · DIMO 2013 · Page 5 (26)

General assumptions f = f 1 + f 2 , g = g 1 + g 2 , where f 2 , g 2 ∈ C 3 [ 0, 1 ] , and where (A2) f 1 , g 1 ∈ C 3 ( 0, 1 ) are convex and satisfy: r ց 0 f ′ r ց 0 g ′ r ր 1 f ′ r ր 1 g ′ 1 ( r ) = lim 1 ( r ) = − ∞ , 1 ( r ) = lim 1 ( r ) = + ∞ lim lim (6) | f ′ 1 ( r ) | ≤ M 1 + M 2 | g ′ ∃ M 1 ≥ 0, M 2 > 0 : 1 ( r ) | ∀ r ∈ ( 0, 1 ) . (7) y 0 Γ = y 0 | Γ , f 1 ( y 0 ) ∈ L 1 ( Ω ) , g 1 ( y 0 Γ ) ∈ L 1 ( Γ ) , and y 0 ∈ V , (A3) 0 < y 0 < 1 a.e. in Ω , 0 < y 0 Γ < 1 a.e. on Γ . (8) U ⊂ X is open such that U ad ∈ U , and there is some R > 0 with (A4) � u � L ∞ ( Q ) + � u Γ � L ∞ ( Σ ) ≤ R ∀ ( u , u Γ ) ∈ U . (9) Optimal control ... · DIMO 2013 · Page 5 (26)

General assumptions Remarks: 1. (A2) implies that the singularity on the boundary grows at least with the same order as the one in the bulk. Optimal control ... · DIMO 2013 · Page 6 (26)

General assumptions Remarks: 1. (A2) implies that the singularity on the boundary grows at least with the same order as the one in the bulk. 2. Typical nonlinearities satisfying (5) and (6) are f 1 ( r ) = c 1 ( r log ( r ) + ( 1 − r ) log ( 1 − r )) , g 1 ( r ) = c 2 ( r log ( r ) + ( 1 − r ) log ( 1 − r )) , where c 1 > 0 , c 2 > 0 . Optimal control ... · DIMO 2013 · Page 6 (26)

General assumptions Remarks: 1. (A2) implies that the singularity on the boundary grows at least with the same order as the one in the bulk. 2. Typical nonlinearities satisfying (5) and (6) are f 1 ( r ) = c 1 ( r log ( r ) + ( 1 − r ) log ( 1 − r )) , g 1 ( r ) = c 2 ( r log ( r ) + ( 1 − r ) log ( 1 − r )) , where c 1 > 0 , c 2 > 0 . 3. We assume here a differentiable situation. The results are submitted to SIAM J. Control Optimization. A non-differentiable case is studied in Colli–Farshbaf-Shaker–Sprekels (in preparation): there, we assume that f 1 = g 1 = I [ 0,1 ] , so that we have to replace f ′ 1 , g ′ 1 in (1) and (2) by the subdifferential ∂ I [ 0,1 ] . Optimal control ... · DIMO 2013 · Page 6 (26)

The state system The following result is a special case of results proved in Calatroni–Colli (Nonlinear Anal. 2013): Theorem 1: Suppose that (A2) , (A3) are satisfied. Then we have: (i) The state system (1)–(3) has for any ( u , u Γ ) ∈ H a unique solution ( y , y Γ ) ∈ Y such that 0 < y < 1 a.e. in Q , 0 < y Γ < 1 a.e. on Σ . (10) (ii) If also (A4) holds, ∃ K ∗ 1 > 0 : for any ( u , u Γ ) ∈ U the associated solution ( y , y Γ ) ∈ Y satisfies � ( y , y Γ ) � Y ≤ K ∗ � f ′ ( y ) � L 2 ( Q ) + � g ′ ( y Γ ) � L 2 ( Σ ) ≤ K ∗ 1 , 1 . (11) Moreover, ∃ K ∗ 2 > 0 : whenever ( u 1 , u 1 Γ ) , ( u 2 , u 2 Γ ) ∈ U are given, then we have � y 1 − y 2 � C 0 ([ 0, T ] ; H ) + �∇ ( y 1 − y 2 ) � L 2 ( Q ) + � y 1 Γ − y 2 Γ � C 0 ([ 0, T ] ; H Γ ) + �∇ Γ ( y 1 Γ − y 2 Γ ) � L 2 ( Σ ) � � ≤ K ∗ � u 1 − u 2 � L 2 ( Q ) + � u 1 Γ − u 2 Γ � L 2 ( Σ ) . (12) 2 Optimal control ... · DIMO 2013 · Page 7 (26)

The state system Remark: 4. Owing to Theorem 1 , the control-to-state mapping S : ( u , u Γ ) �→ S ( u , u Γ ) : = ( y , y Γ ) is defined as a mapping from H into Y . Moreover, S is Lipschitz continuous when viewed as a mapping from the subset U of H into the space � � × � � C 0 ([ 0, T ] ; H ) ∩ L 2 ( 0, T ; V ) C 0 ([ 0, T ] ; H Γ ) ∩ L 2 ( 0, T ; V Γ ) . We now come to a linearized version of Theorem 1 , which will play a central role in the derivation of first-order necessary and second-order sufficient conditions for (CP) . Optimal control ... · DIMO 2013 · Page 8 (26)

A linear system Theorem 2: Let ( u , u Γ ) ∈ H , c 1 ∈ L ∞ ( Q ) , c 2 ∈ L ∞ ( Σ ) , as well as ( w 0 , w 0 Γ ) ∈ V × V Γ with w 0 | Γ = w 0 Γ be given. Then we have: (i) The linear IBVP w t − ∆ w + c 1 ( x , t ) w = u a.e. in Q , (13) ∂ t w Γ − ∆ Γ w Γ + ∂ n w + c 2 ( x , t ) w Γ = u Γ , w | Γ = w Γ , a.e. on Σ , (14) w ( · , 0 ) = w 0 w Γ ( · , 0 ) = w 0 Γ a.e. in Ω , a.e. on Γ , (15) has a unique solution ( w , w Γ ) ∈ Y . (ii) There is some � C > 0 such that: whenever w 0 = 0 and w 0 Γ = 0 then � ( w , w Γ ) � Y ≤ � C � ( u , u Γ ) � H . (16) Optimal control ... · DIMO 2013 · Page 9 (26)

A linear system Idea of Proof: (i) is more or less a consequence of Theorem 1 . Now let w 0 = 0 , w 0 Γ = 0 . Testing (13) by w t and applying Young’s and Gronwall’s inequalities, we easily find � w � H 1 ( 0, T ; H ) ∩ C 0 ([ 0, T ] ; V ) + � w Γ � H 1 ( 0, T ; H Γ ) ∩ C 0 ([ 0, T ] ; V Γ ) ≤ C 1 � ( u , u Γ ) � H . Comparison in (13) yields � ∆ w � L 2 ( Q ) ≤ C 2 � ( u , u Γ ) � H . Then, applying a standard embedding result, � w � L 2 ( 0, T ; H 3/2 ( Ω )) ≤ C 3 � ( u , u Γ ) � H , whence, by the trace theorem, � ∂ n w � L 2 ( 0, T ; H Γ ) ≤ C 4 � ( u , u Γ ) � H . Optimal control ... · DIMO 2013 · Page 10 (26)

A linear system But then, by comparison in (14), � ∆ Γ w Γ � L 2 ( Σ ) ≤ C 5 � ( u , u Γ ) � H , whence � w Γ � L 2 ( 0, T ; H 2 ( Γ )) ≤ C 6 � ( u , u Γ ) � H . Standard elliptic estimates then yield � w � L 2 ( 0, T ; H 2 ( Ω )) ≤ C 7 � ( u , u Γ ) � H . Remark: 5. It cannot be expected that ( w , w Γ ) ∈ L ∞ ( Q ) × L ∞ ( Σ ) , in general. Optimal control ... · DIMO 2013 · Page 11 (26)

An L ∞ bound for ( y , y Γ ) It holds y 0 ∈ L ∞ ( Ω ) , y 0 Γ ∈ L ∞ ( Γ ) , as well as (A5) 0 < ess inf y 0 ( x ) , ess sup y 0 ( x ) < 1, x ∈ Ω x ∈ Ω y 0 Γ ( x ) , y 0 Γ ( x ) < 1 . 0 < ess inf ess sup x ∈ Γ x ∈ Γ Optimal control ... · DIMO 2013 · Page 12 (26)