In´ egalit´ es spectrales pour le contrˆ ole des EDP lin´ eaires : groupe de Schr¨ odinger contre semigroupe de la chaleur. Luc Miller Universit´ e Paris Ouest Nanterre La D´ efense, France Pde’s, Dispersion, Scattering theory and Control theory , Monastir, June 13, 2013. D’apr` es une collaboration avec Thomas Duyckaerts (Univ. Paris 13) : Resolvent conditions for the control of parabolic equations , Journal of Functional Analysis 263 (2012), pp. 3641-3673. http://hal.archives-ouvertes.fr/hal-00620870 Luc Miller, Paris Ouest, France 1 / 20
Outline Part 1: Background on the interior control of linear PDEs 1 Part 2: Resolvent conditions for parabolic equations 2 Part 3: The harmonic oscillator observed from a half-line 3 Part 4: The Lebeau-Robbiano strategy and logarithmic improvements 4 Luc Miller, Paris Ouest, France 2 / 20
Control of the temperature f in a smooth domain M ⊂ R d (Dirichlet), from a chosen source u acting in an open subset Ω ⊂ M during a time T . M Ω Fast null-controllability The heat O.D.E. in E = L 2 ( M ) with input u ∈ L 2 ( R ; E ): ∂ t f − ∆ f = Ω u . ∀ T > 0 , ∀ f (0) ∈ E , ∃ u , f ( T ) = 0 Part 1: Background on the interior control of linear PDEs Luc Miller, Paris Ouest, France 3 / 20
Control of the temperature f in a smooth domain M ⊂ R d (Dirichlet), from a chosen source u acting in an open subset Ω ⊂ M during a time T . M Ω Fast null-controllability (at cost κ T ) The heat O.D.E. in E = L 2 ( M ) with input u ∈ L 2 ( R ; E ): ∂ t f − ∆ f = Ω u . � T � u ( t ) � 2 dt � κ T � f (0) � 2 . ∀ T > 0 , ∀ f (0) ∈ E , ∃ u , f ( T ) = 0 and 0 � Fast final-observability (at cost κ T ) � T � e T ∆ v � 2 � κ T � Ω e t ∆ v � 2 dt , ( FinalObs ) v ∈ E , T > 0 . 0 Part 1: Background on the interior control of linear PDEs Luc Miller, Paris Ouest, France 3 / 20
Links between heat/Schr¨ odinger/waves controllability ∆ is the Laplacian on a bounded M with Dirichlet boundary conditions. Controllabilty of Restriction on Ω Restriction on T Heat eq. ∂ t f − ∆ f = Ω u No No Schr¨ odinger eq. i ∂ t ψ − ∆ ψ = Ω u Yes No ∂ 2 Wave eq. t w − ∆ w = Ω u Yes Yes Part 1: Background on the interior control of linear PDEs Luc Miller, Paris Ouest, France 4 / 20
Links between heat/Schr¨ odinger/waves controllability ∆ is the Laplacian on a bounded M with Dirichlet boundary conditions. Controllabilty of Restriction on Ω Restriction on T Heat eq. ∂ t f − ∆ f = Ω u No No Schr¨ odinger eq. i ∂ t ψ − ∆ ψ = Ω u Yes No ∂ 2 Wave eq. t w − ∆ w = Ω u Yes Yes 1 ∃ T , wave control ⇒ ∀ T , heat control (by the control transmutation method, cf. Russell, Phung, Miller). 2 ∃ T , wave control ⇒ ∀ T , Schr¨ odinger control (by resolvent conditions, cf. Liu, Miller, Tucsnak-Weiss). √ 3 ∃ T , wave control ⇔ ∃ T , wave group control: i ˙ ψ + − ∆ ψ = Ω u (by resolvent conditions, cf. Miller’12) odinger control ⇒ heat control ? This leads to the new question : Schr¨ Part 1: Background on the interior control of linear PDEs Luc Miller, Paris Ouest, France 4 / 20
Links between heat/Schr¨ odinger/waves controllability ∆ is the Laplacian on a bounded M with Dirichlet boundary conditions. Controllabilty of Restriction on Ω Restriction on T Heat eq. ∂ t f − ∆ f = Ω u No No Schr¨ odinger eq. i ∂ t ψ − ∆ ψ = Ω u Yes No ∂ 2 Wave eq. t w − ∆ w = Ω u Yes Yes 1 ∃ T , wave control ⇒ ∀ T , heat control (by the control transmutation method, cf. Russell, Phung, Miller). 2 ∃ T , wave control ⇒ ∀ T , Schr¨ odinger control (by resolvent conditions, cf. Liu, Miller, Tucsnak-Weiss). √ 3 ∃ T , wave control ⇔ ∃ T , wave group control: i ˙ ψ + − ∆ ψ = Ω u (by resolvent conditions, cf. Miller’12) odinger control ⇒ heat control ? This leads to the new question : Schr¨ odinger ⇒ fractional diffusion ∂ t f + ( − ∆) s f = Ω u , s > 1. No but: Schr¨ Part 1: Background on the interior control of linear PDEs Luc Miller, Paris Ouest, France 4 / 20
Abstract semigroup framework: t �→ e − tA observed by C . Hilbert spaces E (states), F (observations). Semigroup e − tA on E . Bounded (in this talk) operator C ∈ L ( E , F ) (defines what is observed). Its adjoint C ∗ defines how the input u : t �→ F acts in order to control. Example (Heat on the domain M observed on Ω ⊂ M ) A = − ∆ � 0, E = F = L 2 ( M ), D ( A ) = H 2 ( M ) ∩ H 1 0 ( M ), C = Ω. Fast null-controllability of ∂ t f + A ∗ f = C ∗ u , with u ∈ L 2 ( R ; F ) � T � u ( t ) � 2 dt � κ T � f (0) � 2 . ∀ T > 0 , ∀ f (0) ∈ E , ∃ u , f ( T ) = 0 and 0 � Fast final-observability (at cost κ T ) � T � e − TA v � 2 � κ T � Ce − tA v � 2 dt , ( FinalObs ) v ∈ E , T > 0 . 0 Part 1: Background on the interior control of linear PDEs Luc Miller, Paris Ouest, France 5 / 20
Resolvent conditions for control (= Hautus tests) From spectral to dynamic inequalities by (unitary) Fourier transform on E . uss’84 test for exponential stability of t �→ e − tA Recall: Huang-Pr¨ � ( A − λ ) − 1 � � m , Re λ < 0 . Part 2: Resolvent conditions for parabolic equations Luc Miller, Paris Ouest, France 6 / 20
Resolvent conditions for control (= Hautus tests) From spectral to dynamic inequalities by (unitary) Fourier transform on E . uss’84 test for exponential stability of t �→ e − tA Recall: Huang-Pr¨ � ( A − λ ) − 1 � � m , Re λ < 0 . Hautus test for observability of t �→ e itA , A = A ∗ , by C , for some T � v � 2 � m � ( A − λ ) v � 2 + ˜ m � Cv � 2 , v ∈ D ( A ) , λ ∈ R . Zhou-Yamamoto’97 (Huang-Pr¨ uss). Burq-Zworski’04 ( ⇒ ). Miller’05 ( ⇔ ): T > π √ m mT / ( T 2 − m π 2 ) . and κ T = 2 ˜ Recall: Observability of t �→ e itA , A = A ∗ , by C for some T means � T � v � 2 � κ T � Ce itA v � 2 dt , ( ExactObs ) v ∈ E . 0 Part 2: Resolvent conditions for parabolic equations Luc Miller, Paris Ouest, France 6 / 20
Resolvent conditions for control (= Hautus tests) From spectral to dynamic inequalities by (unitary) Fourier transform on E . uss’84 test for exponential stability of t �→ e − tA Recall: Huang-Pr¨ � ( A − λ ) − 1 � � m , Re λ < 0 . Hautus test for observability of t �→ e itA , A = A ∗ , by C , for some T � v � 2 � m � ( A − λ ) v � 2 + ˜ m � Cv � 2 , v ∈ D ( A ) , λ ∈ R . Zhou-Yamamoto’97 (Huang-Pr¨ uss). Burq-Zworski’04 ( ⇒ ). Miller’05 ( ⇔ ): T > π √ m mT / ( T 2 − m π 2 ) . and κ T = 2 ˜ Similar Hautus test for wave ¨ w + Aw = C ∗ f , A > 0, for some T � v � 2 � m λ � ( A − λ ) v � 2 + ˜ m � Cv � 2 , v ∈ D ( A ) , λ ∈ R ∗ . Liu’97 (Huang-Pr¨ uss), Miller’05 ( ⇐ ), R.T.T.Tucsnak’05, Miller’12. Part 2: Resolvent conditions for parabolic equations Luc Miller, Paris Ouest, France 6 / 20
Sufficient resolvent conditions for t �→ e − tA , A > 0 odinger ˙ Recall ( ExactObs ) for Schr¨ ψ − iA ψ = 0 ⇒ ( Res ) with δ = 1 ⇒ ( Res ) with δ = 0 ⇔ ( ExactObs ) for wave ¨ w + Aw = 0. Theorem (Duyckaerts-Miller’11: Main Result) If the resolvent condition with power-law factor : ∃ m > 0 , � 1 � λ � ( A − λ ) v � 2 + � Cv � 2 � v � 2 � m λ δ ( Res ) , v ∈ D ( A ) , λ > 0 , holds for some δ ∈ [0 , 1) , then observability ( FinalObs ) holds for all T > 0 with the control cost estimate κ T � ce c / T β for β = 1+ δ 1 − δ and some c > 0 . Here C is bounded, or admissible to some degree (cf. our paper). Part 2: Resolvent conditions for parabolic equations Luc Miller, Paris Ouest, France 7 / 20
Sufficient resolvent conditions for t �→ e − tA , A > 0 odinger ˙ Recall ( ExactObs ) for Schr¨ ψ − iA ψ = 0 ⇒ ( Res ) with δ = 1 ⇒ ( Res ) with δ = 0 ⇔ ( ExactObs ) for wave ¨ w + Aw = 0. Theorem (Duyckaerts-Miller’11: Main Result) If the resolvent condition with power-law factor : ∃ m > 0 , � 1 � λ � ( A − λ ) v � 2 + � Cv � 2 � v � 2 � m λ δ ( Res ) , v ∈ D ( A ) , λ > 0 , holds for some δ ∈ [0 , 1) , then observability ( FinalObs ) holds for all T > 0 with the control cost estimate κ T � ce c / T β for β = 1+ δ 1 − δ and some c > 0 . Here C is bounded, or admissible to some degree (cf. our paper). Theorem (Duyckaerts-Miller’11: Schr¨ odinger to heat) odinger t �→ e itA holds for some T, If ( ExactObs ) for Schr¨ then ( FinalObs ) for “higher-order” heat t �→ e − tA γ , γ > 1 holds for all T. Part 2: Resolvent conditions for parabolic equations Luc Miller, Paris Ouest, France 7 / 20
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