On Controllability of Waves and Geometric Carleman Estimates Arick - PowerPoint PPT Presentation

. . . . . . . . . . . . . . On Controllability of Waves and Geometric Carleman Estimates Arick Shao Queen Mary University of London LMS Hyperbolic Network Meeting Loughborough University 4 March, 2019 Arick Shao (QMUL) Controllability of Waves 4 March, 2019 . . . . . . . . . . . . . . . . . . . . . . . . . . 1 / 39

. Outline . . . . . . . . . . 1. Main problem: Controllability of wave equations. . Problem statement. Survey of methods. 2. New results: Control on time-dependent domains. New observability and Carleman estimates. Lorentzian geometric ideas. 3. Discussion of Carleman estimates. Ideas of proof. Current/future work. Arick Shao (QMUL) Controllability of Waves 4 March, 2019 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 / 39

. Wave Equations . . . . . . . . . Controllability of Waves Background Recall. Wave operators: . Types of wave equations: Free waves Linear waves Nonlinear waves Geometric waves Here, we focus on linear (and free) waves. Waves present in many equations of physics: Maxwell, Yang-Mills, Einstein, Euler equations. Arick Shao (QMUL) Controllability of Waves 4 March, 2019 . . . . . . . . . . . . . . . . 3 / 39 . . . . . . . . . . . . . □ := − ∂ 2 t + ∆ x , L := □ + ∇ X + V . □ φ = 0 L φ = F L φ = G ( φ, ∇ φ ) □ ⇒ □ g := g αβ ∇ αβ

. Given: . . . . . . . . Controllability of Waves Background Solving Wave Equations Problem (Initial-boundary value problem; IBVP) Initial/fjnal times . Spatial domain Unknown: Solve: Wave equation Initial data Boundary data or Theorem (In appropriate function spaces.) Arick Shao (QMUL) Controllability of Waves 4 March, 2019 . . . . . . . . . . . . . . . . 4 / 39 . . . . . . . . . . . . . . T − < T + Ω ⊆ R n open, bounded φ : [ T − , T + ] × ¯ Ω → R L φ | ( T − , T + ) × Ω = F ( φ, ∂ t φ ) | t = T − = ( φ − 0 , φ − 1 ) φ | ( T − , T + ) × ∂Ω = φ d ∂ v φ | ( T − , T + ) × ∂Ω = φ n IBVP has unique solution φ .

. . . . . . . . . . . . Controllability of Waves . Problem Statement Let’s Be More Proactive Question (Version 1) Can we steer solution to desired state? Question (Version 2) Unfortunately, we do not have superpowers. In practice, can only directly afgect part of system. Arick Shao (QMUL) Controllability of Waves 4 March, 2019 . . . . . . . . . . . . . . . . . . . . . . 5 / 39 . . . . . Can we control what happens to φ ? φ as model for physical phenomenon. From initial state ( φ, ∂ t φ ) | t = T − = ( φ − 0 , φ − 1 ) ... ...achieve fjnal state ( φ, ∂ t φ ) | t = T + = ( φ + 0 , φ + 1 ) .

. . . . . . . . . . . . . . . Controllability of Waves Problem Statement Interior and Boundary Control Here, we focus on Dirichlet boundary control. Other problems treated similarly. Arick Shao (QMUL) Controllability of Waves 4 March, 2019 . . . . . . . . . . . . . . . . . . . . . 6 / 39 . . . . Interior control: Steer φ through forcing term. L φ = χ ω G , ω ⊆ Ω . Free to set G on ( T − , T + ) × ω . Boundary control: Steer φ through boundary data. L φ = 0, Γ ⊆ ( T − , T + ) × ∂Ω . Dirichlet: φ | ( T − , T + ) × ∂Ω = χ Γ φ d . Neumann: ∂ ν φ | ( T − , T + ) × ∂Ω = χ Γ φ n .

. . . . . . . . . . . . . . Controllability of Waves Problem Statement Exact Controllability Problem (Exact Dirichlet boundary controllability) Given: Control region Q. Can solutions be controlled via Dirichlet boundary data? Arick Shao (QMUL) Controllability of Waves 4 March, 2019 . . . . . . . . . . . . . . . . . . . . . 7 / 39 . . . . . Γ ⊆ ( T − , T + ) × ∂Ω Goal: ∀ initial/fjnal data ( φ ± 0 , φ ± 1 ) ∈ L 2 ( Ω ) × H − 1 ( Ω ) : Find boundary data φ d ∈ L 2 ( Γ ) , such that... ...solution φ of IBVP (with initial and boundary data)... ...satisfjes ( φ, ∂ t φ ) | t = T + = ( φ + 0 , φ + 1 ) .

. . . . . . . . . . . . . . Controllability of Waves Problem Statement Basic Observations Null controllability. (From time-reversibility of wave equation.) Remark. Contrast with heat equations: Observation. Finite speed of propagation. Arick Shao (QMUL) Controllability of Waves 4 March, 2019 . . . . . . . . . . . . . . 8 / 39 . . . . . . . . . . . . Observation. Can take ( φ + 0 , φ + 1 ) = 0 or ( φ − 0 , φ − 1 ) = 0. From now on, assume ( φ − 0 , φ − 1 ) = 0. Null controllability ̸ ⇔ exact controllability. Information from ∂Ω needs time to travel to all of Ω . Fundamental lower bound for T + − T − .

. . . . . . . . . . . . . . Controllability of Waves Survey of Methods The Adjoint Problem (P1) Consider linear operator Goal: Show S has full range. (P2) Adjoint of S : H 1 Arick Shao (QMUL) Controllability of Waves 4 March, 2019 . . . . . . . . . . . . . . . . . . . . . . 9 / 39 . . . . L φ | ( T − , T + ) × Ω = 0, S : L 2 ( Γ ) → L 2 ( Ω ) × H − 1 ( Ω ) , ( φ, ∂ t φ ) | t = T − = 0, S ( φ d ) = ( φ, ∂ t φ ) | t = T + . φ | ( T − , T + ) × ∂Ω = φ d . L 2 × H − 1 : transposition solutions. L ∗ ψ | ( T − , T + ) × Ω = 0, S ∗ : L 2 ( Ω ) × H 1 0 ( Ω ) → L 2 ( Γ ) , ( ψ, ∂ t ψ ) | t = T + = ( ψ + 0 , ψ + 1 ) , S ∗ ( ψ + 1 , ψ + 0 ) = ∂ ν ψ | Γ . ψ | ( T − , T + ) × ∂Ω = 0. 0 × L 2 : weak solutions.

. . . . . . . . . . . . . . Controllability of Waves Survey of Methods Observability and Controllability (Dolecki–Russell) By duality and the closed range theorem: Main goal: prove observability inequality. (J.-L. Lions) Hilbert uniqueness method (HUM) Arick Shao (QMUL) Controllability of Waves 4 March, 2019 . . . . . . . . . . . . . . . . . . . . . . . . . . 10 / 39 Null/exact controllability ⇔ S is surjective ⇔ ∥ ξ ∥ ≲ ∥ S ∗ ξ ∥ . Last statement known as observability inequality (for (P2) and L ∗ ): ∥ ( ψ, ∂ t ψ )( T + ) ∥ H 1 × L 2 ≲ ∥ ∂ ν ψ ∥ L 2 ( Γ ) . Modern machinery for (observability ⇔ controllability). Methods for generating control φ d (e.g. as minimiser of functional). Can fjnd φ d minimising L 2 ( Γ ) -norm.

. Survey of Methods . . . . . . . . . . Controllability of Waves Methods for Observability I . I. Fourier series methods Apply variants of Ingham’s inequality... ...to Fourier representation of solution. Theorem ( Ingham ) n n 2 dt. Arick Shao (QMUL) Controllability of Waves 4 March, 2019 . . . . . . . . . . . . . . . 11 / 39 . . . . . . . . . . . . . Applies to one spatial dimension: (− ∂ 2 t + ∂ 2 x + α ) ψ = 0. Consider sequence ( λ n ) n ∈ Z of real numbers, with λ n + 1 − λ n ≥ γ > 0 , n ∈ Z . If T > π γ , then for any sequence ( c n ) n ∈ Z , we have ∫ T � � ∑ ∑ | c n | 2 ≲ T ,γ � � c n e i λ n t � � � � − T � �

. . . . . . . . . . . . . . Controllability of Waves Survey of Methods Methods of Observability II II. Multiplier methods Idea. Integrate by parts the RHS of Theorem ( Ho, Lions, ... ) Arick Shao (QMUL) Controllability of Waves 4 March, 2019 . . . . . . . . . . . . . . 12 / 39 . . . . . . . . . . . . Applies to all dimensions: □ ψ = 0 (and small perturbations) ∫ 0 = □ ψ S ∗ ψ . [ T − , T + ] × Ω Fix x 0 ∈ R n , and assume T + − T − > 2 sup | x − x 0 | . x ∈ ∂Ω Then, for L := □ , observability holds with: Γ := ( T − , T + ) × { x ∈ ∂Ω | ( x − x 0 ) · ν > 0 } . Note. ( t , x ) ∈ Γ ⇔ − → x 0 x is leaving Ω at x.

. . . . . . . . . . . . . . Controllability of Waves Survey of Methods Methods for Observability III III. Carleman estimates Weighted (spacetime) integral estimates, with free real parameter. Theorem ( Lasiecka–Triggiani–Zhang, Zhang, ... ) Arick Shao (QMUL) Controllability of Waves 4 March, 2019 . . . . . . . . . . . . . . . . . . . . . 13 / 39 . . . . . Robust technique—extends multiplier methods to general L . Extends to some geometric wave equations on R × M . ∥ e λ F ∇ t , x ψ ∥ 2 L 2 + ∥ e λ F ψ ∥ 2 L 2 ≲ λ − 2 ∥ e λ F □ ψ ∥ 2 λ ≫ 1. L 2 + . . . , Take λ large ⇒ absorb lower-order terms. Previous theorem holds for general L . But, for technical reasons, also requires x 0 ̸∈ ¯ Ω .

. . . . . . . . . . . . Controllability of Waves . Survey of Methods Methods for Observability IV IV. Microlocal methods Most precise, optimal (w.r.t. control region) results. For time-independent (or time-analytic) equations. Theorem ( Bardos–Lebeau–Rauch, Burq, ... ) Bicharacteristics refmect ofg boundary via geometric optics. Arick Shao (QMUL) Controllability of Waves 4 March, 2019 . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 / 39 Extends to geometric wave equations on R × M . Observability on Γ ⇔ geometric control condition. Each bicharacteristic (null geodesic) in [ T − , T + ] × Ω hits Γ . (Le Rousseau–Lebeau–Terpolilli–Trélat) Time-dependent Γ .