A spectral approach of the null controllability of coupled non ...

Spectral properties of A Parabolic controllability properties An applicationA spectral approach of the null controllability ofcoupled non-selfadjoint parabolic systemsMatthieu LéautaudUPMC, Laboratoire Jacques-Louis Lions; Laboratoire POEMS (INRIA)leautaud@ann.jussieu.frjuly 21, 2009

Spectral properties of A Parabolic controllability properties An applicationThe null-controllability problem{∂t u + Au = Bgu |t=0 = u 0(1)• A elliptic• B boundedQuestion: for a fixed T > 0, ∃? a control function g s.t.u |t=T = 0?

Spectral properties of A Parabolic controllability properties An applicationThe null-controllability problem{∂t u + Au = Bgu |t=0 = u 0(1)• A elliptic• B boundedQuestion: for a fixed T > 0, ∃? a control function g s.t.u |t=T = 0?• Scalar parabolic equations: A = −∆ + b · ∇ + c and B = 1( ) ( ) ωu1 −∆ + a b• Parabolic systems: u = A =,u 2 c −∆ + d( ) 0B =1 ω

Spectral properties of A Parabolic controllability properties An applicationTwo different approaches for scalar parabolic equations• Lebeau-Robbiano ’95A = −∆, selfadjoint !local elliptic Carleman estimates; spectral method• Fursikov-Imanuvilov ’96A = −∆ + b(x, t) · ∇ + c(x, t)global parabolic Carleman estimates

Spectral properties of A Parabolic controllability properties An applicationAdvantages of the Lebeau-Robbiano methodThis method• only relies on elliptic Carleman estimates → simpler toproduce• enlights some fundamental properties of the elliptic operatorA: spectral inequality (Lebeau-Zuazua ’98, Jerison-Lebeau’99)• yields the cost of the null-controllability of the low frequenciesof the elliptic operator A• gives an explicit iterative construction of the control function

Spectral properties of A Parabolic controllability properties An applicationOutlineSpectral properties of ASpectral theory of perturbed selfadjoint operators ASpectral inequality for AParabolic controllability propertiesPartial controlFinal controlAn application

Spectral properties of A Parabolic controllability properties An applicationSpectral properties of A: localization of the spectrumPropositionLet A = A 0 + A 1 be an operator on H. Assume that(a) Re(Au, u) H ≥ λ 0 ‖u‖ 2 Hfor all u ∈ D(A),(b) A 0 : D(A 0 ) ⊂ H → H is selfadjoint, positive, with a densedomain and compact resolvent,(c) A 1 : D(A 1 ) ⊂ H → H satisfies|(A 1 u, u) H | ≤ C‖A 1/20u‖ 2qH ‖u‖2−2q Hfor q ∈ [0, 1)

Spectral properties of A Parabolic controllability properties An applicationSpectral properties of A: localization of the spectrumPropositionLet A = A 0 + A 1 be an operator on H. Assume that(a) Re(Au, u) H ≥ λ 0 ‖u‖ 2 Hfor all u ∈ D(A),(b) A 0 : D(A 0 ) ⊂ H → H is selfadjoint, positive, with a densedomain and compact resolvent,(c) A 1 : D(A 1 ) ⊂ H → H satisfies|(A 1 u, u) H | ≤ C‖A 1/20u‖ 2qH ‖u‖2−2q Hfor q ∈ [0, 1)We then have,(i) D(A) = D(A 0 ) ⊂ D(A 1 ) ⊂ H and A has a dense domain anda compact resolvent,(ii) Sp(A) ⊂ P q K 0= {z ∈ C, Re(z) ≥ 0, | Im(z)| < K 0 Re(z) q }(iii) ‖R A (z)‖ L(H) ≤2d(z,Sp(A 0 )) ≤ 2d(z,R +) for z ∈ C \ Pq K 0,

Spectral properties of A Parabolic controllability properties An applicationSpectral properties of A: localization of the spectrumIm(z) ∂P q K 0λ 0Re(z)Figure: Localization of the spectrum of A

Spectral properties of A Parabolic controllability properties An applicationSpectral properties of A: resolvent estimateTheorem (Markus, Katsnel’son, Matsaev, Agranovich,Dzhanlatyan... 1970-2000...)Let 0 < λ 1 ≤ λ 2 ≤ · · · ≤ λ k ≤ · · · be the spectrum of A 0(selfadjoint). Suppose that ∃p > 0 s.t. lim sup j→∞ λ j j −p > 0.Then, setting β = max{0, p −1 − (1 − q)} there exists a sequence0 < α 0 ≤ α 1 ≤ · · · ≤ α k ≤ · · · → ∞ such that‖R A (z)‖ L(H) ≤ e Cα k β , k ∈ N, Re(z) = α k . (2)Example:If A = −∆ + b · ∇ + c, then q = 1/2, p = 2/n (Weyl). Thusβ = max{0, (n − 1)/2}

Spectral properties of A Parabolic controllability properties An applicationSpectral properties of A: resolvent estimateIm(z) ∂P q K 0Re(z)α 0 λ 0 α 1α k−1α kFigure: Resolvent estimate A

Spectral properties of A Parabolic controllability properties An applicationSpectral projectors of ADefinitionSpectral projectorsΠ α = 1 ∫R A (z)dz,2iπ Γ αand Π ∗ α its adjointHolomorphic calculus on Π α H:‖f (A)Π α ‖ L(H) = 1∫2π ∥ f (z)R A (z)dz∥ ≤ Cαe Cαβ sup |f (z)|Γ α L(H) z∈Γ α

Spectral properties of A Parabolic controllability properties An applicationSpectral properties of A: spectral projectorsIm(z) ∂P q K 0Re(z)αα 0α 1λ 0Γ αα k−1 α kFigure: Complex contours around the spectrum of A

Spectral properties of A Parabolic controllability properties An applicationSpectral inequality for ANotation:• H = state space; Y = observation (control) space• B ∗ ∈ L(H, Y ) a bounded observation operator• θ = max{1/2, p −1 − (1 − q)}

Spectral properties of A Parabolic controllability properties An applicationNotation:Spectral inequality for A• H = state space; Y = observation (control) space• B ∗ ∈ L(H, Y ) a bounded observation operator• θ = max{1/2, p −1 − (1 − q)}TheoremSuppose that ∀v ∈ H 2 (0, T 0 ) satisfying v(0) = 0, we have forsome ν ∈ (0, 1]‖v‖ H 1 (ζ,T 0 −ζ) ≤ C‖v‖ 1−νH 1 (0,T 0 )(‖B ∗ ∂ t v(0)‖ Y + ‖(−∂ 2 t + A ∗ )v‖ H 0 (0,T 0 )) ν.Then, ∃C, D > 0 such that ∀α > 0, ∀w ∈ Π ∗ αH,‖w‖ H ≤ Ce Dαθ ‖B ∗ w‖ Y.If A = −∆ + b · ∇ + c, then θ = max{1/2, (n − 1)/2}

Spectral properties of A Parabolic controllability properties An applicationFind g α s.t.⎧⎨⎩Partial control on Π α H∂ t u + Au = Π α Bg αu |t=0 = u 0 ∈ Π α Hu |t=T = 0Theorem∀T > 0, ∃g α ∈ L 2 (0, T ; Y ) driving u 0 to 0 in time T with a costgiven by ‖g α ‖ L 2 (0,T ;Y ) ≤ CT −1/2 e Dαθ ‖u 0 ‖ H .

Spectral properties of A Parabolic controllability properties An applicationFind g α s.t.⎧⎨⎩Partial control on Π α H∂ t u + Au = Π α Bg αu |t=0 = u 0 ∈ Π α Hu |t=T = 0Theorem∀T > 0, ∃g α ∈ L 2 (0, T ; Y ) driving u 0 to 0 in time T with a costgiven by ‖g α ‖ L 2 (0,T ;Y ) ≤ CT −1/2 e Dαθ ‖u 0 ‖ H .Proof. { −∂t w + AAdjoint system:∗ w = 0w |t=T = w T ∈ Π ∗ αH.Observation inequality (spectral inequality):∫ T∫ TT ‖w(0)‖ 2 H ≤ ‖w(t)‖ 2 H dt ≤ C 2 e 2Dαθ ‖B ∗ w(t)‖ 2 Y dt00

Spectral properties of A Parabolic controllability properties An applicationFinal controlTheoremSuppose that θ < 1. Then, for every T > 0, u 0 ∈ H, there exists acontrol function g ∈ L 2 (0, T ; Y ) such that the solution u of theproblem{∂t u + Au = Bg,satisfies u(T ) = 0.u |t=0 = u 0 ∈ H,

Spectral properties of A Parabolic controllability properties An applicationFinal controlTheoremSuppose that θ < 1. Then, for every T > 0, u 0 ∈ H, there exists acontrol function g ∈ L 2 (0, T ; Y ) such that the solution u of theproblem{∂t u + Au = Bg,satisfies u(T ) = 0.u |t=0 = u 0 ∈ H,Idea of the proof (Lebeau-Robbiano ’95): [0, T ] = ⋃ j∈N [a j, a j+1 ],a j+1 = a j + 2T j• if t ∈ (a j , a j + T j ]: g =partial control → cost e Dαθ j• if t ∈ (a j + T j , a j+1 ]: g = 0, parabolic dissipation e −Cα j

Spectral properties of A Parabolic controllability properties An applicationFinal controlIm(z) ∂P q K 0Re(z)αα 0α 1λ 0Γ αα k−1 α kFigure: Elimination and dissipation

Spectral properties of A Parabolic controllability properties An applicationApplication to parabolic systems⎧⎪⎨⎪⎩• A =∂ t u 1 − ∆u 1 + au 1 + bu 2 = 0 in (0, T ) × Ω,∂ t u 2 − ∆u 2 + cu 1 + du 2 = 1 ω g in (0, T ) × Ω,u 1|t=0 = u1 0 , u 2|t=0 = u2 0 in Ω,u 1 = u 2 = 0on (0, T ) × ∂Ω,( −∆ 00 −∆) ( a b+c d) ( ) 0, B =1 ω• p = n/2, q = 0, and θ = max{1/2; n/2 − 1}(θ < 1 ⇔ n ≤ 3)

Spectral properties of A Parabolic controllability properties An applicationApplication to parabolic systemsPropositionSuppose |b(x)| ≥ b 0 > 0 on O and O ∩ ω ≠ ∅Then, ∃C > 0, ϕ ∈ C ∞ 0 (0, T 0) and ν ∈ (0, 1) such that∀v ∈ ( H 2 ((0, T 0 ) × Ω) ) 2 , v|(0,T0 )×∂Ω = 0, we have‖v‖ (H 1 ((ζ,T 0 −ζ)×Ω)) 2 ≤ C‖v‖1−ν (H 1 ((0,T 0 )×Ω)) 2(‖ϕB ∗ v‖ L 2 (Ω) + ‖(−∂ 2 t + A ∗ )v‖ (L 2 ((0,T 0 )×Ω)) 2 ) ν.

Spectral properties of A Parabolic controllability properties An applicationApplication to parabolic systemsPropositionSuppose |b(x)| ≥ b 0 > 0 on O and O ∩ ω ≠ ∅Then, ∃C > 0, ϕ ∈ C ∞ 0 (0, T 0) and ν ∈ (0, 1) such that∀v ∈ ( H 2 ((0, T 0 ) × Ω) ) 2 , v|(0,T0 )×∂Ω = 0, we have‖v‖ (H 1 ((ζ,T 0 −ζ)×Ω)) 2 ≤ C‖v‖1−ν (H 1 ((0,T 0 )×Ω)) 2(‖ϕB ∗ v‖ L 2 (Ω) + ‖(−∂ 2 t + A ∗ )v‖ (L 2 ((0,T 0 )×Ω)) 2 ) ν.• A spectral inequality holds• The partial controllability result holds• In the case n ≤ 3, the controllability result holds

Spectral properties of A Parabolic controllability properties An applicationOther applications...• Fractional parabolic equations{∂t u + A ν u = Bgu |t=0 = u 0 ∈ H.Null-controllable in any time T > 0 in the case ν > θ.• Nodal sets of sums of root functions ofA = −∆ + b · ∇ + cThere exist positive constants C 1 , C 2 such that for all α > 0and ϕ ∈ Π α L 2 (Ω),{ 1H n−1 ({ϕ = 0}) ≤ C 1 α θ + C 2 , θ = max2 ; n − 1 }.2

Spectral properties of A Parabolic controllability properties An applicationConclusions and perspectivesConclusions:• Spectral inequalities for second order elliptic non-selfadjointoperators• Cost of the null-controllability of the low frequencies of theelliptic operator A• Applications to parabolic coupled systems, to nodal sets ofsums of root functions of the elliptic operator APerspectives:• Improve θ ?• Boundary controllability for systems (elliptic estimates andconstruction of the control function)