Optimal controls for gradient systems associated with grain boundary motions

Transcription

1 Optimal controls for gradient systems associated with grain boundary motions Speaker: Shirakawa, Ken (Chiba Univ., Japan) Based on jointworks with: Yamazaki, Noriaki (Kanagawa Univ., Japan) Kenmochi, Nobuyuki (ICM Warsaw, Poland) Watanabe, Hiroshi (Oita Univ., Japan) Moll, Salvador (Univ. Valencia, Spain) INdAM meeting OCERTO 016 Optimal Control for Evolutionary PDEs and Related Topics, Palazzone, Cortona, Italy, June 0-4, 016.

2 0. Content of the talk 1. State system associated with grain boundary motions The model of parabolic PDEs proposed by [Kobayashi Warren Carter](1999) There is only few uniqueness result for the state system, e.g. Yamazaki](008): the result in 1D-case of the spatial domain In 1D case, we can show an example of the no-uniqueness situation [Ito Kenmochi The 1st Main Theorem 1 is concerned with the key-properties of the 1D-state system, and the well-posedness of the approximating systems (with the uniqueness). Optimal control problems Notions of admissibilities: Gilardi Sprekels] (014)) for the controls; for the solutions (cf. [Colli F.-Shaker Setting of the optimal control problems, involving the state system with no-uniqueness (cf. [Kadoya Kenmochi Murase] (010)) The nd Main Theorem is concerned with the existence of optimal controls The 3rd Main Theorem 3 is concerned with the association between the original optimal control problem and its approximating versions 1

3 1. State systems associated with grain boundary motions System (S) ν with ν 0: 0 < T <, := (0, 1), Γ := = {0, 1} t (u Lη) xu = f(t, x), (t, x) Q := (0, T ), t η κ xη + I [0,1] (η) (η 1 ) + α (η) Dθ = u(t, x), (t, x) Q, ( α 0 (η) t θ x α(η) Dθ ) Dθ + ν xθ = 0, in Q, ( 1) k+1 x u + n 0 (u f Γ (t, k)) = 0, (t, k) Σ := (0, T ) Γ, x η = 0 on Σ, θ(t, 0) = 0 and θ(t, 1) = θ Γ, u(0, x) = u 0 (x), η(0, x) = η 0 (x), θ(0, x) = θ 0 (x), x. u = u(t, x): relative temperature, η = η(t, x): orientation degree in a polycrystal, θ = θ(t, x): orientation angle, 0 η(t, x) 1, (t, x) Q I [0,1] : subdifferential of the indicator function I [0,1] on [0, 1]; L > 0, n 0 > 0, κ > 0, θ Γ (0, π): given constants; u 0 L (), η 0 L (), θ 0 L (): initial data.

4 1. State systems associated with grain boundary motions System (S) ν with ν 0: 0 < T <, := (0, 1), Γ := = {0, 1} t (u Lη) xu = f(t, x), (t, x) Q := (0, T ), t η κ xη + I [0,1] (η) (η 1 ) + α (η) Dθ = u(t, x), (t, x) Q, ( α 0 (η) t θ x α(η) Dθ ) Dθ + ν xθ = 0, in Q, ( 1) k+1 x u + n 0 (u f Γ (t, k)) = 0, (t, k) Σ := (0, T ) Γ, x η = 0 on Σ, θ(t, 0) = 0 and θ(t, 1) = θ Γ, u(0, x) = u 0 (x), η(0, x) = η 0 (x), θ(0, x) = θ 0 (x), x. f L (Q), f Γ L (Σ): heat sources (controls). α 0 = α 0 (η) 0: mobility (possibly degenerate); α = α(η) > 0: mobility, with the differential α (no-degenerate). Typical choice (cf. [Kobayashi Warren Carter](1999)): α 0 (η) = α(η) = η /, η R

5 Non-isothermal system (S) ν (coupling system of Fix-Caginalp type): t (u Lη) xu = f in Q, [ ] t η (B.C.)+(I.C.) = F α 0 (η) t θ ν (u, η, θ) in Q, Free-energy (ν > 0) [Warren Kobayashi Lobkovsky Carter](003) v = [u, η, θ] L () 3 F ν (v) := A 0 u dx + Ψ [0,1] (η) 1 ( η 1 ) dx + Φν (η; θ) (, ]. A 0 > 0: const. (depending on L and n 0 ); η H 1 () Ψ [0,1] (η) := κ x η dx + I [0,1] (η) dx; η C(), θ H 1 () Φ ν (η; θ) := α(η) x θ dx + ν subject to the boundary condition θ(0) = 0 and θ(1) = θ Γ. x θ dx, 3

6 Non-isothermal system (S) ν (coupling system of Fix-Caginalp type): t (u Lη) xu = f in Q, [ ] t η (B.C.)+(I.C.) = F α 0 (η) t θ ν (u, η, θ) in Q, Free-energy (ν = 0) [Warren Kobayashi Lobkovsky Carter](003) v = [u, η, θ] L () 3 F 0 (v) := A 0 u dx + Ψ [0,1] (η) 1 ( η 1 ) dx + Φ0 (η; θ) (, ]. A 0 > 0: const. (depending on L and n 0 ); η H 1 () Ψ [0,1] (η) := κ x η dx + η C(), θ BV () Φ 0 (η; θ) := I [0,1] (η) dx; α(η) D[θ] ex, with the extension [θ] ex BV loc (R) s.t. [θ] ex 0 on (, 0] and [θ] ex θ Γ on [1, ). 3

7 Approximating system (S) ν ε with ν 0 and ε (0, 1 ]: t (u Lη) xu = f(t, x), (t, x) Q, t η κ xη + β ε (η) (η 1 ) + α (η) x θ + ε = u(t, x), (t, x) Q, ) x θ α ε (η) t θ x (α(η) x θ + ε + (ν + ε) xθ = 0, in Q, (B.C.) + (I.C.) α ε = α 0 + ε; β ε : Yosida resularization of I [0,1] ; Approximating free-energy: v = [u, η, θ] L () 3 Fε ν (v) := A 0 u dx + Ψ [0,1] (η) 1 ( η 1 ) dx + Φ ν ε (η; θ) (, ]. θ H 1 () Φ ν ε(η; θ) := α(η) x θ + ε dx + ν + ε x θ dx, subject to the boundary condition θ(0) = 0 and θ(1) = θ Γ, η C(). 4

8 Assumptions. (A1) 0 α 0 C 1 (R ), s.t. α0 1 (0) = {0}. (A) 0 < α C (R): convex, α (0) = 0, and δ := inf α(r) > 0. Notations V := H 1 (): Hilbert space, endowed with the inner product: (v, z) V := ( x v, x z) L () + n 0 (v, z) L (Γ), [v, z] V. V : dual space of V, F : V V duality map. Range for controls: H := L (Q) L (Σ). We regard any f = [f, f Γ ] H, as f V, via the identification: f, z = (f, z) L () + n 0 (f Γ, z) L (Γ), z V. Range for solutions: { D ν := ṽ = [ũ, η, θ] L () 3 η H 1 (), θ D(Φ ν ( η; )) 0 η 1 and 0 θ θ Γ a.e. in } 5

9 Definition of solution to (S) ν : ν 0, f = [f, f Γ ] H, v 0 = [u 0, η 0, θ 0 ] D ν, v = [u, η, θ] L (Q) 3 is called a solution to (S) ν, iff. the following conditions hold. (S0) u W 1, (0, T ; V ) L (0, T ; L ()) L (0, T ; V ), u(0) = u 0 in L (), η W 1, (0, T ; L ()) L (0, T ; H 1 ()) C(Q), η(0) = η 0 in L (), α 0 (η)θ W 1, (0, T ; L ()), Φ ν (η; θ) L (0, T ), α 0 (η)θ(0) = α(η 0 )θ 0 in L (). θ := t [α 0 (η)θ] t [α 0 (η)]θ L (Q), θ = α 0 (η) t θ in Q \ η 1 (0). (S1) u solves the following evolution equation: t ( u Lη ) (t) + F u(t) = f(t) in V, a.e. t (0, T ). (S) η solves the following variational inequality: ( t η(t) (η(t) 1 ), η(t) φ) L () + Ψ [0,1] (η(t)) Ψ [0,1] (φ) + (η(t) φ)α (η(t)) D[θ(t)] ex 0, φ H 1 (), a.e. t (0, T ). (S3) θ solves the following evolution equation: θ (t) + Φ ν (η(t); θ(t)) 0, in L (), a.e. t (0, T ), where η H 1 (), Φ ν ( η; ) denotes L -subdifferential of Φ ν ( η; ). 6

10 Definition of solution to (S) ν ε : ν 0, ε (0, 1 ], f = [f, f Γ] H, v 0 = [u 0, η 0, θ 0 ] D ν, v = [u, η, θ] L (Q) 3 is called a solution to (S) ν ε, iff. the following conditions hold. (S0) ε u W 1, (0, T ; V ) L (0, T ; L ()) L (0, T ; V ), u(0) = u 0 in L (), η W 1, (0, T ; L ()) L (0, T ; H 1 ()) C(Q), η(0) = η 0 in L (), θ W 1, (0, T ; L ()) L (0, T ; H 1 ()) C(Q), (S1) ε u solves the following evolution equation: θ(0) = θ 0 in L (). t ( u Lη ) (t) + F u(t) = f(t) in V, a.e. t (0, T ). (S) ε η solves the following variational identity: ( t η(t) (η(t) 1 ), φ) L () + κ( x η(t), x φ) L () + (β ε (η(t)), φ) L () +(α (η(t)) x θ(t) + ε, φ) L () = 0, φ H 1 (), a.e. t (0, T ). (S3) ε θ solves the following evolution equation: α ε (η(t)) t θ(t) + Φ ν ε(η(t); θ(t)) 0, in L (), a.e. t (0, T ), where η H 1 (), Φ ν ε( η; ) denotes L -subdifferential of Φ ν ( η; ). 7

11 Main Theorem 1 (Key-properties of solutions). (I) ν 0, let us define: S ν :H D ν [f, v 0 ] { v = [u, η, θ] : solution to (S) ν } L (Q) 3. Then, [f, v 0 ] H D ν, S ν (f, v 0 ), NOT singleton in general, and moreover: { νn ν, f n f in H, v 0,n v 0 in L () 3, as n, {F νn (v 0,n )}: b.d.d., and v n = [u n, η n, θ n ] S νn (f n, v 0,n ), n N, = {n k } {n}, v = [u, η, θ] S ν (f, v 0 ), s.t. v nk = [u nk, η nk, θ nk ] v = [u, η, θ] weakly- in L (0, T ; L ()), [u nk, η nk, α 0 (η nk )θ nk ] [u, η, α 0 (η)θ] in L (Q) C(Q) C([0, T ]; L ()), νnk θ nk νθ in L (0, T ; H 1 ()), as k. (II) ν 0, ε (0, 1 ], let us define: S ν ε :H D ν [f, v 0 ] { v = [u, η, θ] : solution to (S) ν ε} L (Q) 3. Then, [f, v 0 ] H D ν, S ν ε (f, v 0 ) be a singleton and continuous w.r.t. the strong topologies from H L () 3 into L (Q) C(Q) C([0, T ]; L ()).. (I) is obtained by the observation of approximating limit ε 0, based on (II). 8

12 Main Theorem 1 (Key-properties of solutions). (I) ν 0, let us define: S ν :H D ν [f, v 0 ] { v = [u, η, θ] : solution to (S) ν } L (Q) 3. Then, [f, v 0 ] H D ν, S ν (f, v 0 ), NOT singleton in general, and moreover: { νn ν, f n f in H, v 0,n v 0 in L () 3, as n, {F νn (v 0,n )}: b.d.d., and v n = [u n, η n, θ n ] S νn (f n, v 0,n ), n N, = {n k } {n}, v = [u, η, θ] S ν (f, v 0 ), s.t. v nk = [u nk, η nk, θ nk ] v = [u, η, θ] weakly- in L (0, T ; L ()), [u nk, η nk, α 0 (η nk )θ nk ] [u, η, α 0 (η)θ] in L (Q) C(Q) C([0, T ]; L ()), νnk θ nk νθ in L (0, T ; H 1 ()), as k. (II) ν 0, ε (0, 1 ], let us define: S ν ε :H D ν [f, v 0 ] { v = [u, η, θ] : solution to (S) ν ε} L (Q) 3. Then, [f, v 0 ] H D ν, S ν ε (f, v 0 ) be a singleton and continuous w.r.t. the strong topologies from H L () 3 into L (Q) C(Q) C([0, T ]; L ()).. (II) is deduced from [Kenmochi Niezgódka](1994), [Ito Kenmochi Yamazaki(008)]. 8

13 Example of no-uniqueness situation: when ν = 0 and f = [f, f Γ ] [0, 0] Multiple solutions v = [ū, η, θ] with common initial data: ū(t, x) = 0 and η(t, x) = 1 ( sin ( x 1 κ ) + 1 ) χ( 1 π κ, 1+π κ ) (x) + χ ( 1+π κ,1) (x), (t, x) Q steady-state in Fix Caginalp system [Chen Elliott](1994) 9

14 Example of no-uniqueness situation: when ν = 0 and f = [f, f Γ ] [0, 0] Multiple solutions v = [ū, η, θ] with common initial data: θ(t, x) = b(t)χ (0,ā(t)) (x) + θ Γ χ (ā(t),1) (x), ā(t) η 1 (0), b(t) [0, θ Γ ], (t, x) Q = α 0 ( η) t θ(t) + Φ0 ( η(t); θ(t)) 0 in L (), a.e. t (0, T ) 9

15 . Optimal controls problems Keypoint: admissible classes ν 0, v 0 = [u 0, η 0, θ 0 ] D ν : fixed Class of admissible controls: M > 0, H M := { f = [f, f Γ ] H f H 1 (Q) M and f Γ H 1 (Σ) M }. H M is to give a constraint for the controls, to get the compactness for in H (cf. [Colli F.-Shaker Gilardi Sprekels] (014)). Class of admissible solutions: M > 0, f H M, {v n } = {Sε ν n (f n, v 0,n )} L (Q) 3 with {ε n < n } (0, 1 ], {f n} H M, {v 0,n D ν+εn } L () 3, s.t. A ν M(f, v 0 ) := v S ν (f, v 0 ) v n v weakly in L (Q) 3, f n f in H, v 0,n v 0 in L () 3, F ν ε n (v 0,n ) F ν (v 0 ), as n. In the light of Main Theorem 1, the weak convergence for {v n } can be changed by: [u n, η n, α εn (η n )θ n ] [u, η, α 0 (η)θ] in L (Q) C(Q) C([0, T ]; L ()), as n. 10

16 Target profile: v = [u, η, θ ] L (Q) 3 Optimal control problem (OP; v 0 ) ν M with ν 0, M > 0, v 0 D ν : to find f = [f, f Γ ] H M (optimal control), which minimize the following cost functional J M on H M : f = [f, f Γ ] H M J M (f) := inf { ϖ f (v) v A ν M (f, v 0) }, with v = [u, η, θ] L (Q) L (Q) ϖ f (v) := 1 + (η η )(t) L () + ˆα 0 (η)θ α 0 (η )θ (t) + 1 Z T 0 ` f(t) L () + f Γ(t) L (Γ) dt Z T 0 L () dt. This setting is refer to [Kadoya Kenmochi Murase] (010). ` (u u )(t) L () + Main Theorem (Existence of optimal controls). ν 0, M > 0, v 0 D ν, [f, v ] (optimal pair), s.t.: d M(v 0 ) := inf J M (f) = ϖ f (v ) f H M 11

17 Target profile: v = [u, η, θ ] L (Q) 3 Approximating control problem (OP; v 0 ) ν M,ε with ν, M, ε (0, 1 ], v 0,ε D ν+ε : to find f ε = [f ε, f Γ,ε ] H M (approximating optimal control), which minimize the following approximating cost functional J M,ε on H M : f H M J M,ε (f) := 1 T 0 ( (u u )(t) L () + + (η η )(t) L () + [ α ε (η)θ α ε (η )θ ] (t) L ()) dt + 1 T 0 ( f(t) L () + f Γ(t) L (Γ)) dt, with v = [u, η, θ] := S ν ε (f, v 0 ) Proposition 1 (Existence of approximating optimal controls). ν 0, M > 0, ε (0, 1 ], v 0,ε D ν+ε, [f ε, v ε] (approximating optimal pair), s.t.: d M,ε(v 0,ε ) := inf J M,ε (f) f H M. This proposition is obtained by means of the standard argument of minimizing sequence. 1

18 Main Theorem 3 (Association between the control problems) Let ν 0, M > 0 be fixed. Then, v 0 D ν, [f, v ] H M A ν M (f; v 0) (optimal pair), {ε n < n }, {v 0,n D ν+εn }, {[f n, v n]} H M S ν ε n (f n; v 0,n ) (sequence of approximating optimal pair) s.t. : d M,ε n (v 0,n ) = J M.εn (f n) d M(v 0 ) = ϖ f (v ) as n Keypoints of the proofs Main Theorem : diagonal argument Let us take: a minimizing sequence {f n } H M for the cost functional J M, with a sequence of admissible solutions v n A ν M (f n, v 0 ), n N = {ε m < m }, {v 0,m D ν+εm }, {f n m} H M with {v n m = Sν εm (f n, v 0 )}, s.t. : [ ] v 0,m v 0 in L () 3, Fε ν n (v 0,m ) F ν (v 0 ), Admissibility f n m f n in H, as m, n N of solutions = Applying the diagonal argument: [f, v ] H M A ν M (f, v 0 ) (optimal pair) [ Admissibility of controls ] 13

19 Main Theorem 3: a consequence from Main Theorem 1 & the admissibilities Let us take: any optimal pair [ f, v] H M A ν M ( f, v 0 ) = { ε n < n }, { v 0,n D ν+ εn }, { f n } H M with { v n = S ν εn ( f n, v 0,n )}, s.t. : [ ] Main Theorem 1 d M(v 0 ) = lim J M, ε n ( f n ) lim n n d M, ε n ( v 0,n ) & Admissibility of solutions On the other hand: taking approximating optimal pairs [ f n, v n] H M S ν ε n ( f n, v 0,n ), [ ] Main Theorem 1 lim d M, ε n ( v 0,n ) = lim ϖ n n f ( v n n) d M(v 0 ) & Admissibility of controls Remark. If ν > 0, then the conclusions as in Main Theorems 3 can be obtained by adopting the following standard type functional: [f, v] H L (Q) 3 1 v v L (Q) + 1 f H, with the target v L (Q) 3 as a constitute conponent of the cost functional. 14

20 3. Problems in future ( I ) Necessary conditions for the optimal pairs Strategy : Necessary condition for the approximating optimal pairs Observation for the original optimal pairs, via the approximating limit ( II ) Expansion of the results to the general higher-dimenaional cases Keypoints : to find an appropriate approximating state systems, which have the both of uniqueness and smoothness (III) Expansion of the results to the cases with anisotropies Strategy : continuation of the recent study [Moll S. Watanabe](010) concerned with the anisotropic versions of the state systems (IV) Structural observations for solutions Keypoints : previous works of total-variation flows, e.g. [Andreu Caselles Mazón](004), [Bellettini Caselles Novaga](00), [Kobayashi Giga](1999), [Giga Giga](010), [Moll](005 ), [Rybka Mucha](000 ), [S.](000 ), e.t.c. Strategy : to obtain some additional properties of solutions, including the concrete profiles of optimal pairs (V) Uniqueness of the state systems Status : there is no advance, yet 15