Lecture 4: Some numerical methods for control problems Enrique - PowerPoint PPT Presentation

Lecture 4: Some numerical methods for control problems Enrique FERNÁNDEZ-CARA Dpto. E.D.A.N. - Univ. of Sevilla Numerical controllability The 1D heat equation The Navier-Stokes system Other parabolic PDEs and systems E. Fernández-Cara Numerics in control problems

Outline Background 1 Controllability - the 1D heat equation Controllability - the Navier-Stokes system The heat equation: numerical results 2 The Fursikov-Imanuvilov formulation Direct finite element approximation The numerical controllability of a nonlinear heat equation 3 Problems and results E. Fernández-Cara Numerics in control problems

Controllability problems, examples and applications Examples and applications FIRST (SIMPLE) EXAMPLE: 1D heat: y t − y xx = v 1 ω , ( x , t ) ∈ ( 0 , 1 ) × ( 0 , T )   y ( 0 , t ) = y ( 1 , t ) = 0 , t ∈ ( 0 , T ) (H) y ( x , 0 ) = y 0 ( x ) , x ∈ ( 0 , 1 )  We assume: ω = ( a , b ) , 0 < a < b < 1 Null controllability problem: For all y 0 find v such that y ( T ) = 0 NC? Yes, for all ω and T Applications: Heating and cooling, controlling a population, etc. E. Fernández-Cara Numerics in control problems

Controllability problems, examples and applications Examples and applications A numerical experiment Ω = ( 0 , 1 ) , ω = ( 0 . 2 , 0 . 8 ) , T = 0 . 5, y 0 ( x ) ≡ sin( π x ) , y t − ay xx = v 1 ω , a = 10 − 1 1 0 � 1 � 2 � 3 � 4 � 5 1 0.8 0.5 0.6 0.4 0.3 0.4 x 0.2 0.2 0.1 t 0 0 Figure: ω = ( 0 . 2 , 0 . 8 ) . The control E. Fernández-Cara Numerics in control problems

Controllability problems, examples and applications Examples and applications 1 0.8 0.6 0.4 0.2 0 1 0.8 0.5 0.6 0.4 0.3 0.4 x 0.2 0.2 t 0.1 0 0 Figure: ω = ( 0 . 2 , 0 . 8 ) . The state E. Fernández-Cara Numerics in control problems

Controllability problems, examples and applications Examples and applications SECOND (NOT SO SIMPLE) EXAMPLE: Navier-Stokes: y t + ( y · ∇ ) y − ν 0 ∆ y + ∇ p = v 1 ω , ∇ · y = 0   y ( x , t ) = 0 , ( x , t ) ∈ ∂ Ω × ( 0 , T ) y ( x , 0 ) = y 0 ( x )  AC? NC? ECT? OPEN What we know: Local results Theorem [EFC-Guerrero-Imanuvilov-Puel 2004] Fix a solution ( y , p ) , with y ∈ L ∞ ∃ ε > 0 such that � y 0 − y ( 0 ) � H 1 0 ≤ ε ⇒ ∃ controls such that y ( T ) = y ( T ) E. Fernández-Cara Numerics in control problems

Controllability problems, examples and applications Examples and applications y t + ( y · ∇ ) y − ν 0 ∆ y + ∇ p = v 1 ω , ∇ · y = 0   y ( x , t ) = 0 , ( x , t ) ∈ ∂ Ω × ( 0 , T ) y ( x , 0 ) = y 0 ( x )  ∃ other results, among them: - Global AC for when N = 2, Navier boundary conditions [Coron 1996] - Global boundary “AC” in a 3D cube [Guerrero-Imanuvilov-Puel 2012] - Global NC with periodicity [Fursikov-Imanuvilov 1999], without boundary [Coron-Fursikov 1996], . . . E. Fernández-Cara Numerics in control problems

Controlling fluids Exact controllability to a fixed flow - Numerical approximations and results A numerical experiment: Taylor-Green (vortex) flow y = (sin( 2 x 1 ) cos( 2 x 2 ) e − 8 t , − cos( 2 x 1 ) sin( 2 x 2 ) e − 8 t ) Figure: Taylor-Green flow E. Fernández-Cara Numerics in control problems

Controlling fluids Exact controllability to a fixed flow - Numerical approximations and results A numerical experiment: Taylor-Green (vortex) flow Figure: The Taylor-Green velocity field E. Fernández-Cara Numerics in control problems

Controlling fluids Exact controllability to a fixed flow - Numerical approximations and results A numerical experiment: Taylor-Green (vortex) flow Ω = ( 0 , π ) × ( 0 , π ) , ω = ( π/ 3 , 2 π/ 3 ) × ( 0 , 1 ) , T = 1 y 0 = y + m z , z = ∇ × ψ , ψ = ( π − y ) 2 y 2 ( π − x ) 2 x 2 ( m << 1) Approximation: P 2 in ( x 1 , x 2 ) and t + multipliers . . . – freefem ++ Figure: The mesh − Nodes: 3146, Elements: 15900, Variables: 7 × 3146 E. Fernández-Cara Numerics in control problems

Controlling fluids Exact controllability to a fixed flow - Numerical approximations and results A numerical experiment: Taylor-Green (vortex) flow Figure: The initial state E. Fernández-Cara Numerics in control problems

Controlling fluids Exact controllability to a fixed flow - Numerical approximations and results A numerical experiment: Taylor-Green (vortex) flow Figure: The state at t = 0 . 6 E. Fernández-Cara Numerics in control problems

Controlling fluids Exact controllability to a fixed flow - Numerical approximations and results A numerical experiment: Taylor-Green (vortex) flow Figure: The state at t = 0 . 9 Taylor-Green Vortex.edp E. Fernández-Cara Numerics in control problems

Numerics: methods and results The Fursikov-Imanuvilov strategy for the heat equation The problem: y t − y xx = v 1 ω ( H 1 ) + . . . Goal: “Good” v such that y ( T ) = 0 numerically, i.e. � y ( T ) � L 2 ∼ 10 − 10 Glowinski, JL Lions, Boyer-Hubert-Le Rousseau, Münch . . . The “classical” way: ω × ( 0 , T ) | v | 2 dx dt Minimize �� Subject to ( H 1 ) , y ( T ) = 0 Eventually: penalize the functional and/or the PDEs But: Numericall ill-posed, leads to oscillations! E. Fernández-Cara Numerics in control problems

Numerics: methods and results The Fursikov-Imanuvilov strategy for the heat equation ALTERNATIVE METHOD: introduce ρ, ρ 0 ∼ e C ( x ) / ( T − t ) and solve The weighted (FI) formulation of the NC problem: ( ρ 2 | y | 2 + 1 ω ρ 2 0 | v | 2 ) Minimize �� Subject to ( H 1 ) , y ( T ) = 0 Notation: Ly = y t − y xx , L ∗ p = − p t − p xx Note: y = y + Lv Euler-Lagrange characterization of the optimal ( v , y ) : y = ρ − 2 L ∗ p , v = − ρ − 2 0 p | ω × ( 0 , T ) together with the weak (Lax-Milgram) formulation ( ρ − 2 L ∗ p L ∗ ϕ + ρ − 2 Ω y 0 ( x ) ϕ ( x , 0 ) dx 0 1 ω p ϕ ) dx dt = � �� � ∀ ϕ ∈ P ; p ∈ P m ( p , ϕ ) = � ℓ, ϕ � ∀ ϕ ∈ P ; p ∈ P ( ρ − 2 | L ∗ ϕ | 2 + ρ − 2 0 1 ω | ϕ | 2 ) < + ∞ , ϕ | x = 0 ≡ ϕ | x = 1 ≡ 0 } �� P = { ϕ : E. Fernández-Cara Numerics in control problems

Numerics: methods and results The Fursikov-Imanuvilov strategy for the heat equation Lax-Milgram formulation: m ( p , ϕ ) = � ℓ, ϕ � ∀ ϕ ∈ P ; p ∈ P ( ρ − 2 | L ∗ ϕ | 2 + ρ − 2 0 1 ω | ϕ | 2 ) < + ∞ , ϕ | x = 0 ≡ ϕ | x = 1 ≡ 0 } �� P = { ϕ : A weak formulation of L ( ρ − 2 L ∗ p ) + ρ − 2 0 1 ω p = 0 , ( x , t ) ∈ ( 0 , 1 ) × ( 0 , T )   p ( 0 , t ) = p ( 1 , t ) = 0 , t ∈ ( 0 , T )   ( ρ − 2 L ∗ p ) | x = 0 = ( ρ − 2 L ∗ p ) | x = 1 = 0 , t ∈ ( 0 , T )   ( ρ − 2 L ∗ p ) | t = 0 = y 0 ( x ) , ( ρ − 2 L ∗ p ) | t = T = 0 , x ∈ ( 0 , 1 )  Attention: 2nd order in time, 4th order in space ∃ ! solution in view of Carleman: 2 ( | ϕ t | 2 + | ∆ ϕ | 2 ) + ρ − 2 1 |∇ ϕ | 2 + ρ − 2 ρ − 2 0 | ϕ | 2 � �� � I ( ϕ ) := ( ρ − 2 | L ∗ ϕ | 2 + ρ − 2 0 1 ω | ϕ | 2 ) �� ≤ C ∀ ϕ ∈ P ⇒ The ellipticity of m ( · , · ) in L 2 loc and . . . The continuity of ℓ in P E. Fernández-Cara Numerics in control problems

Numerics: methods and results The Fursikov-Imanuvilov strategy for the heat equation m ( p , ϕ ) = � ℓ, ϕ � ∀ ϕ ∈ P ; p ∈ P ( ρ − 2 | L ∗ ϕ | 2 + ρ − 2 0 1 ω | ϕ | 2 ) < + ∞ , ϕ | x = 0 ≡ ϕ | x = 1 ≡ 0 } �� P = { ϕ : Then: y = ρ − 2 L ∗ p , v = − ρ − 2 0 p 1 ω Standard finite element approximation: m ( p h , ϕ h ) = � ℓ, ϕ h � ∀ ϕ h ∈ P h ; p h ∈ P h Then: ( ρ − 2 | L ∗ ϕ | 2 + ρ − 2 0 1 ω | ϕ | 2 ) < + ∞ , ϕ | x = 0 ≡ ϕ | x = 1 ≡ 0 } �� P h ⊂ P = { ϕ : i.e. necessarily C 1 in x and C 0 in t finite elements (relatively bad news) Alternative: mixed formulation (multipliers) + C 0 in x and t finite elements Standard FEM framework (good news): convergence in P for usual polynomial P h with ∪ h P h = P EFC-Münch, Cindea-EFC-Münch, EFC-Münch-Souza, . . . E. Fernández-Cara Numerics in control problems

Numerics: methods and results The Fursikov-Imanuvilov strategy for the heat equation Coming back to the first numerical experiment Ω = ( 0 , 1 ) , ω = ( 0 . 2 , 0 . 8 ) , T = 0 . 5, y 0 ( x ) ≡ sin( π x ) , y t − ay xx = v 1 ω , a = 10 − 1 Approximation: P 3 , x ⊗ P 1 , t , C 1 in x , C 0 in t 1 0 � 1 � 2 � 3 � 4 � 5 1 0.8 0.5 0.6 0.4 0.3 0.4 x 0.2 0.2 0.1 t 0 0 Figure: ω = ( 0 . 2 , 0 . 8 ) . The control E. Fernández-Cara Numerics in control problems

Numerics: methods and results The Fursikov-Imanuvilov strategy for the heat equation 1 0.8 0.6 0.4 0.2 0 1 0.8 0.5 0.6 0.4 0.3 0.4 0.2 x 0.2 t 0.1 0 0 Figure: ω = ( 0 . 2 , 0 . 8 ) . The state E. Fernández-Cara Numerics in control problems

The numerical controllability of a nonlinear heat equation Problems and results An extension: A semilinear heat equation:  y t − ( a ( x ) y x ) x + f ( y ) = v 1 ω , ( x , t ) ∈ ( 0 , 1 ) × ( 0 , T )  y ( 0 , t ) = 0 , ( x , t ) ∈ { 0 , 1 } × ( 0 , T ) y ( x , 0 ) = y 0 ( x ) , x ∈ ( 0 , 1 )  We assume: f : R �→ R is Lipschitz-continuous, with | f ( s ) | ∼ | s | log p ( 1 + | s | ) as | s | → + ∞ , p ≥ 1 Known results, [EFC-Zuazua, 2000], [Barbu, 2000] - Recall: If f ( 0 ) = 0 and p < 3 / 2, NC holds ∀ p > 2 ∃ f with f ( 0 ) = 0 such that NC does not hold It is unknown what happens when 3 / 2 ≤ p ≤ 2 E. Fernández-Cara Numerics in control problems