Inégalités spectrales pour le contrôle des EDP linéaires : groupe de Schrödinger contre semigroupe de la chaleur.

Inégalités spectrales pour le contrôle des EDP linéaires : groupe de Schrödinger contre semigroupe de la chaleur. Luc Miller Université Paris Ouest Nanterre La Défense, France Pde s, Dispersion, Scattering theory and Control theory, Monastir, June 13, 2013. D après 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 1 Part 1: Background on the interior control of linear PDEs 2 Part 2: Resolvent conditions for parabolic equations 3 Part 3: The harmonic oscillator observed from a half-line 4 Part 4: The Lebeau-Robbiano strategy and logarithmic improvements 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 = Ωu. 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 = Ωu. T > 0, f (0) E, u, f (T ) = 0 and T 0 u(t) 2 dt κ T f (0) 2. Fast final-observability (at cost κ T ) (FinalObs) T e T v 2 κ T Ωe t v 2 dt, 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/schrödinger/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 = Ωu No No Schrödinger eq. i t ψ ψ = Ωu Yes No Wave eq. t 2 w w = Ωu Yes Yes 1 T, wave control T, heat control (by the control transmutation method, cf. Russell, Phung, Miller). 2 T, wave control T, Schrödinger control (by resolvent conditions, cf. Liu, Miller, Tucsnak-Weiss). 3 T, wave control T, wave group control: i ψ + ψ = Ωu (by resolvent conditions, cf. Miller 12) This leads to the new question : Schrödinger control heat control? Part 1: Background on the interior control of linear PDEs Luc Miller, Paris Ouest, France 4 / 20

Links between heat/schrödinger/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 = Ωu No No Schrödinger eq. i t ψ ψ = Ωu Yes No Wave eq. t 2 w w = Ωu Yes Yes 1 T, wave control T, heat control (by the control transmutation method, cf. Russell, Phung, Miller). 2 T, wave control T, Schrödinger control (by resolvent conditions, cf. Liu, Miller, Tucsnak-Weiss). 3 T, wave control T, wave group control: i ψ + ψ = Ωu (by resolvent conditions, cf. Miller 12) This leads to the new question : Schrödinger control heat control? No but: Schrödinger fractional diffusion t f + ( ) s f = Ωu, s > 1. 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) H0 1 (M), C = Ω. Fast null-controllability of t f + A f = C u, with u L 2 (R; F) T > 0, f (0) E, u, f (T ) = 0 and Fast final-observability (at cost κ T ) (FinalObs) T 0 u(t) 2 dt κ T f (0) 2. T e TA v 2 κ T Ce ta v 2 dt, 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. Recall: Huang-Prüss 84 test for exponential stability of t e ta (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-Prüss). Burq-Zworski 04 ( ). Miller 05 ( ): T > π m and κ T = 2 mt /(T 2 mπ 2 ). Recall: Observability of t e ita, A = A, by C for some T means (ExactObs) T v 2 κ T Ce ita v 2 dt, 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. Recall: Huang-Prüss 84 test for exponential stability of t e ta (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-Prüss). Burq-Zworski 04 ( ). Miller 05 ( ): T > π m and κ T = 2 mt /(T 2 mπ 2 ). Similar Hautus test for wave ẅ + 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-Prüss), 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 Recall (ExactObs) for Schrödinger ψ iaψ = 0 (Res) with δ = 1 (Res) with δ = 0 (ExactObs) for wave ẅ + Aw = 0. Theorem (Duyckaerts-Miller 11: Main Result) If the resolvent condition with power-law factor : m > 0, (Res) v 2 mλ δ ( 1 λ (A λ)v 2 + Cv 2 ), 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 Recall (ExactObs) for Schrödinger ψ iaψ = 0 (Res) with δ = 1 (Res) with δ = 0 (ExactObs) for wave ẅ + Aw = 0. Theorem (Duyckaerts-Miller 11: Main Result) If the resolvent condition with power-law factor : m > 0, (Res) v 2 mλ δ ( 1 λ (A λ)v 2 + Cv 2 ), 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: Schrödinger to heat) If (ExactObs) for Schrödinger t e ita holds for some T, then (FinalObs) for higher-order heat t e taγ, γ > 1 holds for all T. Part 2: Resolvent conditions for parabolic equations Luc Miller, Paris Ouest, France 7 / 20

Sufficient resolvent conditions for t e ta, A > 0 log-improvement of λ δ, δ < 1, into λ/(ϕ(λ)) 2, ϕ(λ) = (log λ) α, α > 1. Theorem (Duyckaerts-Miller 11: Main Result, log-improved) If the resolvent condition with logarithmic factor : m > 0, v 2 mλ ( ) 1 (ϕ(λ)) 2 λ (A λ)v 2 + Cv 2, v D(A), λ > 0, holds for some α > 1, then observability (FinalObs) holds for all T > 0. Here C is bounded, or admissible for the wave equation ẅ + Aw = 0. Theorem (Duyckaerts-Miller 11: Schrödinger to heat, log-improved) If (ExactObs) for Schrödinger t e ita holds for some T, then (FinalObs) for higher-order heat t e taϕ(1+a), α > 1, T > 0. Part 2: Resolvent conditions for parabolic equations Luc Miller, Paris Ouest, France 8 / 20

Application to the control of diffusions in a potential well A = + V on E = L 2 (R), D(A) = { u H 2 (R) Vu L 2 (R) }. V (x) = x 2k, k N, k > 0. C = Ω = (, x 0 ), x 0 R. Part 2: Resolvent conditions for parabolic equations Luc Miller, Paris Ouest, France 9 / 20

Application to the control of diffusions in a potential well A = + V on E = L 2 (R), D(A) = { u H 2 (R) Vu L 2 (R) }. V (x) = x 2k, k N, k > 0. C = Ω = (, x 0 ), x 0 R. Theorem (Miller at CPDEA, IHP 10) ( ) 1 v 2 mλ 1/k λ (A λ)v 2 + Cv 2, v D(A), λ > 0, and the decay of the first coefficient cannot be improved. Theorem (Duyckaerts-Miller 11) The diffusion in the potential well V (x) = x 2k, k N, k > 1, t φ 2 x φ V φ = Ωu, φ(0) = φ 0 L 2 (R), u L 2 ([0, T ] R), is null-controllable in any time, i.e. T > 0, φ 0, u such that φ(t ) = 0. Part 2: Resolvent conditions for parabolic equations Luc Miller, Paris Ouest, France 9 / 20

Necessary resolvent conditions for any semigroup t e ta Example (worst resolvent condition for the Laplacian on a manifold) A is the Laplacian on the unit sphere S 2 = { x 2 + y 2 + z 2 = 1 }, C = Ω is the complement a neighborhood of the great circle {z = 0}. e n (x, y, z) = (x + iy) n : (A λ n )e n = 0 and a > 0, e n ae a λ n Ce n. This leads to the resolvent condition with exponential factor : m > 0, (Res) v 2 me m(re λ)α ( (A λ)v 2 + Cv 2), v D(A), Re λ > 0. Theorem (Duyckaerts-Miller 11) If (FinalObs) holds for some T > 0 then (Res) holds with α = 1. If (FinalObs) holds for all T (0, T 0 ] with the control cost κ T = ce c/t β for some β > 0, c > 0, T 0 > 0, then (Res) holds with α = β β+1 < 1. Still valid for C L(D(A), F) admissible, i.e. T 0 Ce ta v 2 dt k T v 2. Part 2: Resolvent conditions for parabolic equations Luc Miller, Paris Ouest, France 10 / 20

The harmonic oscillator observed from a half-line Ω R Disproves : controllability of Schrödinger eq. controllability of heat eq. t φ 2 x φ + x 2 φ = Ωu versus i t ψ 2 x ψ + x 2 ψ = Ωu Here Ω = (, x 0 ), x 0 R, and A = 2 x + x 2 on E = L 2 (R) = F. Theorem (Miller at CPDEA, IHP 10) Observability (FinalObs) for heat t e ta does not hold for any time. Observability (ExactObs) for Schrödinger t e ita holds for some time. Eigenvalues are λ n = 2n + 1. N.b. 1 λ n = + but dim F 1. Eigenfunctions e n are e n (x) = c n ( x x) n e x2 /2 = c n H n (x)e x2 /2, where c n = ( π2 n (n!)) 1/2, H n = ( 1) n e x2 n x e x2 are the Hermite polynomials. Part 3: The harmonic oscillator observed from a half-line Luc Miller, Paris Ouest, France 11 / 20

Sketch of proof 1: non-observability for the heat semigroup Harmonic oscillator A = 2 x + x 2 observed from a half line Ω = (, x 0 ). Disprove (FinalObs) + (e TA v)(x) 2 dx κ 2 T T x0 0 (e ta v)(x) 2 dxdt, by taking the Dirac mass at y / Ω as initial data v and letting y. Part 3: The harmonic oscillator observed from a half-line Luc Miller, Paris Ouest, France 12 / 20

Sketch of proof 1: non-observability for the heat semigroup Harmonic oscillator A = 2 x + x 2 observed from a half line Ω = (, x 0 ). Disprove (FinalObs) + (e TA v)(x) 2 dx κ 2 T T x0 0 (e ta v)(x) 2 dxdt, by taking the Dirac mass at y / Ω as initial data v and letting y. More precisely, v(x) = e εa (x, y), where ε is a small time, e ta (x, y) is the kernel of the operator e ta. Hence (e ta v)(x) = e (t+ε)a (x, y). Bound from below the fundamental state e 0 hence the final state e TA v e (T +ε)λ 0 e 0 (y) c T exp ( y 2 ). 2 Bound from above the kernel hence the observation: Mehler formula e ta e t ( (x, y) = π(1 e 4t )) exp 1 + e 4t x 2 + y 2 ) 1 e 4t + 2e 2t xy. 2 1 e 4t Part 3: The harmonic oscillator observed from a half-line Luc Miller, Paris Ouest, France 12 / 20

Sketch of proof 2: observability for the Schrödinger group Harmonic oscillator A = x 2 + x 2 observed from a half line Ω = (, x 0 ). Prove (Res) using a semiclassical reduction and microlocal propagation. By the change of variable u(y) = v(x), y = hx, h = 1/λ, (Res) v 2 m (A λ)v 2 + m Ωv 2, v D(A), λ > 0, reduces to the semiclassical resolvent condition + u(y) 2 dy m + h 2 h 2 u (y) + (y 2 1)u(y) 2 dy hx 0 + m u(y) 2 dy, u C 0 (R), h (0, 1]. Part 3: The harmonic oscillator observed from a half-line Luc Miller, Paris Ouest, France 13 / 20

Sketch of proof 2: observability for the Schrödinger group Harmonic oscillator A = 2 x + x 2 observed from a half line Ω = (, x 0 ). Prove (Res) using a semiclassical reduction and microlocal propagation. By the change of variable u(y) = v(x), y = hx, h = 1/λ, (Res) v 2 m (A λ)v 2 + m Ωv 2, v D(A), λ > 0, reduces to the semiclassical resolvent condition + u(y) 2 dy m + h 2 h 2 u (y) + (y 2 1)u(y) 2 dy hx 0 + m u(y) 2 dy, u C 0 (R), h (0, 1]. Arguing by contradiction, introduce a semiclassical measure (= Wigner measure) in phase space (x, ξ) R 2 : it is supported on { x 2 + ξ 2 = 1 }, invariant by rotation and supported in {x 0}. Part 3: The harmonic oscillator observed from a half-line Luc Miller, Paris Ouest, France 13 / 20

An observability estimate for sums of eigenfunctions M Ω M v(x) 2 dx ce c λ Ω v(x) 2 dx, for all λ > 0 and v = µ λ e µ, where { e µ = µe µ on M e µ = 0 on M. Lebeau-Robbiano 95 (Carleman estimates), Lebeau-Jerison 96, Lebeau-Zuazua 98. Part 4: The Lebeau-Robbiano strategy and logarithmic improvements Luc Miller, Paris Ouest, France 14 / 20

The direct Lebeau-Robbiano strategy We may write the previous spectral observability estimate concisely with spectral subspaces of the Dirichlet Laplacian E λ = Span µ λ e µ : v ãe a λ Ωv, v E λ, λ > 0. More generally E λ may be defined by some functional calculus. For example, when A is self-adjoint: E λ = 1 A<λ E. Observability on spectral subspaces (with power α (0, 1)) (SpecObs) v ãe aλα Cv, v E λ, λ λ 0 > 0. Fast final-observability (at cost κ T ce c/t β, β = (FinalObs) α ) 1 α T e TA v 2 κ T Ce ta v 2 dt, v E, T > 0. 0 Part 4: The Lebeau-Robbiano strategy and logarithmic improvements Luc Miller, Paris Ouest, France 15 / 20

Dynamic spectral inequality for the direct L.-R. strategy Observability on spectral subspaces (with power α (0, 1)) (SpecObs) v ãe aλα Cv, v E λ, λ λ 0 > 0. β > 0 Dynamic observability on spectral subspaces (α (0, 1)) T e TA v 2 ãe aλα +b/t β Ce ta v 2 dt, v E λ, T > 0, λ λ 0. 0 β = α 1 α Fast final-observability (at cost κ T ce c/t β, β = (FinalObs) α ) 1 α T e TA v 2 κ T Ce ta v 2 dt, v E, T > 0. 0 Part 4: The Lebeau-Robbiano strategy and logarithmic improvements Luc Miller, Paris Ouest, France 16 / 20

Sufficient resolvent conditions for t e ta, A > 0 Now we are ready to sketch the proof of the main result, which we recall: Theorem (Duyckaerts-Miller 11: Main Result) If the resolvent condition with power-law factor : m > 0, (Res) v 2 mλ δ ( 1 λ (A λ)v 2 + Cv 2 ), 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. In this talk, we consider only δ = 1/3 to simplify the computations. Part 4: The Lebeau-Robbiano strategy and logarithmic improvements Luc Miller, Paris Ouest, France 17 / 20

Sketch of proof of the Main Result (with δ = 1/3) Recall A > 0, E λ = 1A<λ E hence E λ 2 = 1 A<λ E. (Res) v 2 mλ 1/3 ( 1 λ (A λ)v 2 + Cv 2 ), v D(A), λ > 0. T (FinalObs) e TA v 2 ce c/t 2 Ce ta v 2 dt, v E, T > 0. 0 Part 4: The Lebeau-Robbiano strategy and logarithmic improvements Luc Miller, Paris Ouest, France 18 / 20

Sketch of proof of the Main Result (with δ = 1/3) Recall A > 0, E λ = 1A<λ E hence E λ 2 = 1 A<λ E. (Res) v 2 mλ 1/3 ( 1 λ (A λ)v 2 + Cv 2 ), v D(A), λ > 0. v 2 mλ 2/3 ( ( A λ)v 2 + Cv 2), v D( A), λ > 0. T (FinalObs) e TA v 2 ce c/t 2 Ce ta v 2 dt, v E, T > 0. 0 Part 4: The Lebeau-Robbiano strategy and logarithmic improvements Luc Miller, Paris Ouest, France 18 / 20

Sketch of proof of the Main Result (with δ = 1/3) Recall A > 0, E λ = 1A<λ E hence E λ 2 = 1 A<λ E. (Res) v 2 mλ 1/3 ( 1 λ (A λ)v 2 + Cv 2 ), v D(A), λ > 0. v 2 mλ 2/3 ( ( A λ)v 2 + Cv 2), v D( A), λ > 0. Controllability of waves on E λ 2 E λ 2 for times λ 1/3 at cost λ 1/3. T (FinalObs) e TA v 2 ce c/t 2 Ce ta v 2 dt, v E, T > 0. 0 Part 4: The Lebeau-Robbiano strategy and logarithmic improvements Luc Miller, Paris Ouest, France 18 / 20

Sketch of proof of the Main Result (with δ = 1/3) Recall A > 0, E λ = 1A<λ E hence E λ 2 = 1 A<λ E. (Res) v 2 mλ 1/3 ( 1 λ (A λ)v 2 + Cv 2 ), v D(A), λ > 0. v 2 mλ 2/3 ( ( A λ)v 2 + Cv 2), v D( A), λ > 0. Controllability of waves on E λ 2 E λ 2 for times λ 1/3 at cost λ 1/3. Controllability of heat on E λ 2 for all T > 0 at cost e λ2/3 /T. T (FinalObs) e TA v 2 ce c/t 2 Ce ta v 2 dt, v E, T > 0. 0 Part 4: The Lebeau-Robbiano strategy and logarithmic improvements Luc Miller, Paris Ouest, France 18 / 20

Sketch of proof of the Main Result (with δ = 1/3) Recall A > 0, E λ = 1A<λ E hence E λ 2 = 1 A<λ E. (Res) v 2 mλ 1/3 ( 1 λ (A λ)v 2 + Cv 2 ), v D(A), λ > 0. v 2 mλ 2/3 ( ( A λ)v 2 + Cv 2), v D( A), λ > 0. Controllability of waves on E λ 2 E λ 2 for times λ 1/3 at cost λ 1/3. Controllability of heat on E λ 2 for all T > 0 at cost e λ2/3 /T. Controllability of heat on E λ for all T > 0 at cost e λ1/3 /T, but λ 1/3 /T λ α + 1/T β where α = 2/3 and β = 2 satisfy β = α 1 α, hence the direct Lebeau-Robbiano strategy in the previous slide applies. T (FinalObs) e TA v 2 ce c/t 2 Ce ta v 2 dt, v E, T > 0. 0 Part 4: The Lebeau-Robbiano strategy and logarithmic improvements Luc Miller, Paris Ouest, France 18 / 20

The direct Lebeau-Robbiano Strategy: log-improvement Here A is self-adjoint, E λ = 1A<λ E, C is bounded or admissible. Theorem (Duyckaerts-Miller 11: logarithmic L.-R. strategy) Logarithmic observability on spectral subspaces with α > 2 v 2 ae aλ/((log(log λ))α log λ) Cv 2, v E λ, λ λ 0 > e. (FinalObs) T e TA v 2 κ T Ce ta v 2 dt, v E, T > 0. 0 Theorem (Duyckaerts-Miller 11: logarithmic anomalous diffusion) Let ϕ(λ) = (log λ) α, α > 1 or ϕ(λ) = (log(log λ)) α log λ, α > 2. The following anomalous diffusion is null-controllable in any time T > 0: t φ + ϕ( )φ = Ωu, φ(0) = φ 0 L 2 (M), u L 2 ([0, T ] M). Part 4: The Lebeau-Robbiano strategy and logarithmic improvements Luc Miller, Paris Ouest, France 19 / 20

Advertisement of selected references Downloads on http://hal.archives-ouvertes.fr/aut/luc+miller/. About the direct Lebeau-Robbiano strategy: On the cost of fast control for heat-like semigroups: spectral inequalities and transmutation, PICOF 10, hal-00459601. A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups, DCDS 10, hal-00411846. Seidman 08. Tenenbaum-Tucsnak 10. About the Hautus test for conservative PDE s: Tucsnak-Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basel Textbooks, 09. Resolvent conditions for the control of unitary groups and their approximations, JST 12, hal-00620772. Ervedoza 08 (approximation). Jacob-Zwart 09 (other semigroups). About the Hautus test for parabolic PDE s: Resolvent conditions for the control of parabolic equations, Joint work with Thomas Duyckaerts, JFA 12, hal-00620870. Part 4: The Lebeau-Robbiano strategy and logarithmic improvements Luc Miller, Paris Ouest, France 20 / 20