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Lecture 4: Some numerical methods for control problems Enrique FERNNDEZ-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. Fernndez-Cara

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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

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SLIDE 2

Outline

1

Background Controllability - the 1D heat equation Controllability - the Navier-Stokes system

2

The heat equation: numerical results The Fursikov-Imanuvilov formulation Direct finite element approximation

3

The numerical controllability of a nonlinear heat equation Problems and results

  • E. Fernández-Cara

Numerics in control problems

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SLIDE 3

Controllability problems, examples and applications

Examples and applications

FIRST (SIMPLE) EXAMPLE: 1D heat: (H)    yt − yxx = v1ω, (x, t) ∈ (0, 1) × (0, T) y(0, t) = y(1, t) = 0, t ∈ (0, T) 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

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SLIDE 4

Controllability problems, examples and applications

Examples and applications

A numerical experiment Ω = (0, 1), ω = (0.2, 0.8), T = 0.5, y0(x) ≡ sin(πx), yt − ayxx = v1ω, a = 10−1

0.1 0.2 0.3 0.4 0.5 0.2 0.4 0.6 0.8 1 5 4 3 2 1 1

t x

Figure: ω = (0.2, 0.8). The control

  • E. Fernández-Cara

Numerics in control problems

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SLIDE 5

Controllability problems, examples and applications

Examples and applications

0.1 0.2 0.3 0.4 0.5 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

t x

Figure: ω = (0.2, 0.8). The state

  • E. Fernández-Cara

Numerics in control problems

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Controllability problems, examples and applications

Examples and applications

SECOND (NOT SO SIMPLE) EXAMPLE: Navier-Stokes:    yt + (y · ∇)y − ν0∆y + ∇p = v1ω, ∇ · 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)H1

0 ≤ ε ⇒ ∃ controls such that y(T) = y(T)

  • E. Fernández-Cara

Numerics in control problems

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SLIDE 7

Controllability problems, examples and applications

Examples and applications

   yt + (y · ∇)y − ν0∆y + ∇p = v1ω, ∇ · 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

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Controlling fluids

Exact controllability to a fixed flow - Numerical approximations and results

A numerical experiment: Taylor-Green (vortex) flow y = (sin(2x1) cos(2x2)e−8t, − cos(2x1) sin(2x2)e−8t)

Figure: Taylor-Green flow

  • E. Fernández-Cara

Numerics in control problems

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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

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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, 2π/3) × (0, 1), T = 1 y0 = y + m z, z = ∇ × ψ, ψ = (π − y)2y 2(π − x)2x2 (m << 1) Approximation: P2 in (x1, x2) and t + multipliers . . . – freefem++

Figure: The mesh − Nodes: 3146, Elements: 15900, Variables: 7×3146

  • E. Fernández-Cara

Numerics in control problems

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SLIDE 11

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

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SLIDE 12

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

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SLIDE 13

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

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Numerics: methods and results

The Fursikov-Imanuvilov strategy for the heat equation

The problem: (H1) yt − yxx = v1ω + . . . Goal: “Good” v such that y(T) = 0 numerically, i.e. y(T)L2 ∼ 10−10 Glowinski, JL Lions, Boyer-Hubert-Le Rousseau, Münch . . . The “classical” way: Minimize

  • ω×(0,T) |v|2 dx dt

Subject to (H1), 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

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Numerics: methods and results

The Fursikov-Imanuvilov strategy for the heat equation

ALTERNATIVE METHOD: introduce ρ, ρ0 ∼ eC(x)/(T−t) and solve The weighted (FI) formulation of the NC problem: Minimize

  • (ρ2|y|2 + 1ωρ2

0|v|2)

Subject to (H1), y(T) = 0 Notation: Ly = yt − yxx, L∗p = −pt − pxx Note: y = y + Lv Euler-Lagrange characterization of the optimal (v, y): y = ρ−2L∗p, v = −ρ−2

0 p|ω×(0,T)

together with the weak (Lax-Milgram) formulation (ρ−2L∗p L∗ϕ + ρ−2

0 1ωp ϕ) dx dt =

  • Ω y 0(x) ϕ(x, 0) dx

∀ϕ ∈ P; p ∈ P m(p, ϕ) = ℓ, ϕ ∀ϕ ∈ P; p ∈ P P = {ϕ :

  • (ρ−2|L∗ϕ|2 + ρ−2

0 1ω|ϕ|2) < +∞, ϕ|x=0 ≡ ϕ|x=1 ≡ 0}

  • E. Fernández-Cara

Numerics in control problems

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SLIDE 16

Numerics: methods and results

The Fursikov-Imanuvilov strategy for the heat equation

Lax-Milgram formulation: m(p, ϕ) = ℓ, ϕ ∀ϕ ∈ P; p ∈ P P = {ϕ :

  • (ρ−2|L∗ϕ|2 + ρ−2

0 1ω|ϕ|2) < +∞, ϕ|x=0 ≡ ϕ|x=1 ≡ 0}

A weak formulation of        L(ρ−2L∗p) + ρ−2

0 1ωp = 0,

(x, t) ∈ (0, 1)×(0, T) p(0, t) = p(1, t) = 0, t ∈ (0, T) (ρ−2L∗p)|x=0 = (ρ−2L∗p)|x=1 = 0, t ∈ (0, T) (ρ−2L∗p)|t=0 = y 0(x), (ρ−2L∗p)|t=T = 0, x ∈ (0, 1) Attention: 2nd order in time, 4th order in space ∃! solution in view of Carleman: I(ϕ) := ρ−2

2 (|ϕt|2 + |∆ϕ|2) + ρ−2 1 |∇ϕ|2 + ρ−2 0 |ϕ|2

≤ C

  • (ρ−2|L∗ϕ|2 + ρ−2

0 1ω|ϕ|2)

∀ϕ ∈ P ⇒ The ellipticity of m(· , ·) in L2

loc and . . .

The continuity of ℓ in P

  • E. Fernández-Cara

Numerics in control problems

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SLIDE 17

Numerics: methods and results

The Fursikov-Imanuvilov strategy for the heat equation

m(p, ϕ) = ℓ, ϕ ∀ϕ ∈ P; p ∈ P P = {ϕ :

  • (ρ−2|L∗ϕ|2 + ρ−2

0 1ω|ϕ|2) < +∞, ϕ|x=0 ≡ ϕ|x=1 ≡ 0}

Then: y = ρ−2L∗p, v = −ρ−2

0 p1ω

Standard finite element approximation: m(ph, ϕh) = ℓ, ϕh ∀ϕh ∈ Ph; ph ∈ Ph Then: Ph ⊂ P = {ϕ :

  • (ρ−2|L∗ϕ|2 + ρ−2

0 1ω|ϕ|2) < +∞, ϕ|x=0 ≡ ϕ|x=1 ≡ 0}

i.e. necessarily C1 in x and C0 in t finite elements (relatively bad news) Alternative: mixed formulation (multipliers) + C0 in x and t finite elements Standard FEM framework (good news): convergence in P for usual polynomial Ph with ∪hPh = P EFC-Münch, Cindea-EFC-Münch, EFC-Münch-Souza, . . .

  • E. Fernández-Cara

Numerics in control problems

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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, y0(x) ≡ sin(πx), yt − ayxx = v1ω, a = 10−1 Approximation: P3,x ⊗ P1,t, C1 in x, C0 in t

0.1 0.2 0.3 0.4 0.5 0.2 0.4 0.6 0.8 1 5 4 3 2 1 1

t x

Figure: ω = (0.2, 0.8). The control

  • E. Fernández-Cara

Numerics in control problems

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Numerics: methods and results

The Fursikov-Imanuvilov strategy for the heat equation

0.1 0.2 0.3 0.4 0.5 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

t x

Figure: ω = (0.2, 0.8). The state

  • E. Fernández-Cara

Numerics in control problems

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The numerical controllability of a nonlinear heat equation

Problems and results

An extension: A semilinear heat equation:    yt − (a(x)yx)x + f(y) = v1ω, (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| logp(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

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The numerical controllability of a nonlinear heat equation

Problems and results

In the sequel: two methods from [EFC-Münch, 2012] 1 - Fixed-point iterates Introduce g, with g(s) = f(s)/s if s = 0, g(0) = f ′(0) y = Λ(z) ⇔ y solves, together with v, the NC problem    yt − (ax)yx)x + g(z)y = v1ω, (x, t) ∈ (0, 1) × (0, T) y(0, t) = 0, (x, t) ∈ {0, 1} × (0, T) y(x, 0) = y 0(x), y(x, T) = 0, x ∈ (0, 1) ALG 1 (Fixed-point) y n+1 = Λ(y n), n ≥ 0

  • E. Fernández-Cara

Numerics in control problems

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The numerical controllability of a nonlinear heat equation

Problems and results

2 - Least-squares and gradient techniques Set R(z) = 1

2z − Λ(z)2 for z ∈ L2(Q)

y = Λ(z) ⇔ y solves, together with v, the NC problem    yt − (a(x)yx)x + g(z)y = v1ω, (x, t) ∈ (0, 1) × (0, T) y(0, t) = 0, (x, t) ∈ {0, 1} × (0, T) y(x, 0) = y 0(x), y(x, T) = 0, x ∈ (0, 1) Least-squares formulation of the NC problem:    Minimize R(z) := 1 2z − Λ(z)2 Subject to z ∈ L2(Q) ALG 2 (Leat-squares + descent) Apply (for instance) the conjugate gradient algorithm to the search of a minimum of R

  • E. Fernández-Cara

Numerics in control problems

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The numerical controllability of a nonlinear heat equation

Problems and results

The NC problems for the linear heat equations solved as before Imanuvilov’s formulation of the NC problem: Introduce weights ρ, ρ0 ∼ eC(x)/(T−t) and formulate a different problem Minimize

  • (ρ2|y|2 + 1ωρ2

0|v|2), subject to v ∈ L2(ω × (0, T)), (v, y) solving

. . . Notation: Ly = yt − (a(x)yx)x + g(z)y, L∗p = −pt − (a(x)px)x + g(z)p Euler-Lagrange characterization of the optimal (v, y): y = ρ−2L∗p, v = −ρ−2

0 p|ω×(0,T)

with        L(ρ−2L∗p) + ρ−2

0 1ωp = 0,

(x, t) ∈ (0, 1)×(0, T) p(0, t) = p(1, t) = 0, t ∈ (0, T) (ρ−2L∗p)|x=0 = (ρ−2L∗p)|x=1 = 0, t ∈ (0, T) (ρ−2L∗p)|t=0 = y 0(x), (ρ−2L∗p)|t=T = 0, x ∈ (0, 1) Remember: 2nd order in time, 4th order in space But can be solved numerically through an appropriate FEM!

  • E. Fernández-Cara

Numerics in control problems

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The numerical controllability of a nonlinear heat equation

Problems and results

A numerical experiment Data: Ω = (0, 1), ω = (0.2, 0.8), T = 0.5, f(y) ≡ −5y log1.4(1 + |y|) y0(x) ≡ 40 sin(πx), FreeFem++ & mesh adaptation Mixed formulation for linear problems, approximation: P2 in (x, t), C0 in (x, t) ρ = 0.1eβ(x)/(T−t) with β ∼ 1, |β′| > 0 outside ω Stopping test: y n+1 − y nL2/y nL2 < 10−7 The initial mesh — Nodes: 7634, Elements: 14906, Variables: 3 × 7634

  • E. Fernández-Cara

Numerics in control problems

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SLIDE 25

The numerical controllability of a nonlinear heat equation

Problems and results

The final mesh — Nodes: 4175, Elements: 8128, Variables: 3 × 4175

  • E. Fernández-Cara

Numerics in control problems

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The numerical controllability of a nonlinear heat equation

Problems and results

The control

  • E. Fernández-Cara

Numerics in control problems

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SLIDE 27

The numerical controllability of a nonlinear heat equation

Problems and results

0,1 0,2 0,3 0,4 0.5 0,25 0,5 0,75 1 2000 1500 1000 500

t x Figure: The control, 3D view

  • E. Fernández-Cara

Numerics in control problems

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The numerical controllability of a nonlinear heat equation

Problems and results

Figure: The state

  • E. Fernández-Cara

Numerics in control problems

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SLIDE 29

The numerical controllability of a nonlinear heat equation

Problems and results

0,1 0,2 0,3 0,4 0,5 0.25 0.5 0.75 1 20 20 40

x t Figure: The state, 3D view

  • E. Fernández-Cara

Numerics in control problems

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THANK YOU VERY MUCH . . .

  • E. Fernández-Cara

Numerics in control problems