Moments and convex optimization for analysis and control of nonlinear - - PowerPoint PPT Presentation

Moments and convex optimization for analysis and control of nonlinear partial differential equations

Milan Milan Kor Korda da

(LAAS-CNRS, Toulouse)

Didier Henrion

(LAAS-CNRS, Toulouse)

Jean-Bernard Lasserre

(LAAS-CNRS, Toulouse)

Problem setting – analysis

Milan Korda 2

F(x, y(x), Dy(x)) = 0 for x ∈ Ωo G(x, y(x), Dy(x)) = 0 for x ∈ ∂Ω,

Ω ⊂ Rn compact

y : Rn → Rk

Problem setting – analysis

Milan Korda 3

Ω ⊂ Rn compact

F(x, y(x), Dy(x)) = 0 for x ∈ Ωo G(x, y(x), Dy(x)) = 0 for x ∈ ∂Ω,

y : Rn → Rk

J

• y(·)
• :=

R

Ω L (x, y(x), Dy(x)) dx +

R

∂Ω L∂ (x, y(x), Dy(x)) dσ(x)

Problem setting – analysis

Milan Korda 4

Ω ⊂ Rn compact

F(x, y(x), Dy(x)) = 0 for x ∈ Ωo G(x, y(x), Dy(x)) = 0 for x ∈ ∂Ω,

y : Rn → Rk

Example: L(x, y, z) = y 2, L∂(x, y, z)

J(y(·)) = R

Ω y(x)2 dx

⇒

J

• y(·)
• :=

R

Ω L (x, y(x), Dy(x)) dx +

R

∂Ω L∂ (x, y(x), Dy(x)) dσ(x)

Problem setting – analysis

Milan Korda 5

Ω ⊂ Rn compact

F(x, y(x), Dy(x)) = 0 for x ∈ Ωo G(x, y(x), Dy(x)) = 0 for x ∈ ∂Ω,

y : Rn → Rk

inf / sup

y

J

• y
• subject to

y solves the PDE

Goal: establish bounds on

J

• y(·)
• :=

R

Ω L (x, y(x), Dy(x)) dx +

R

∂Ω L∂ (x, y(x), Dy(x)) dσ(x)

Infinite dimensional linear programming

Milan Korda 6

Aµ = b

µ ∈ M+

⇒

min

µ hg, µi

Infinite-dimensional LP

ANy = bN

min

y hgN, yi

y ∈ M+

N

⇒

Convex semidefinite program

inf / sup

y

J

• y
• subject to

y solves the PDE

Infinite dimensional linear programming

Milan Korda 7

Aµ = b

µ ∈ M+

⇒

min

µ hg, µi

Infinite-dimensional LP

ANy = bN

min

y hgN, yi

y ∈ M+

N

⇒

Convex semidefinite program

inf / sup

y

J

• y
• subject to

y solves the PDE

Long history: Global optimization (Lasserre, Parrilo, Nesterov,...)

Stability of ODEs (Rantzer, Vaydia,...)

Optimal control of ODEs (Young, Vinter, Lasserre, Henrion, Gaitsgory,...)

. . .

Occupation and boundary measures

Milan Korda 8

Occupation and boundary measures

Milan Korda 9

µ(A × B × C) =

• Ω

IA×B×C (x, y(x), Dy(x)) dx

Occupation measure

Occupation and boundary measures

Milan Korda 10

µ(A × B × C) =

• Ω

IA×B×C (x, y(x), Dy(x)) dx

Occupation measure

x

y(x)

µ(A × B)

A × B

Occupation and boundary measures

Milan Korda 11

µ(A × B × C) =

• Ω

IA×B×C (x, y(x), Dy(x)) dx

µ∂(A × B × C) =

• ∂Ω

IA×B×C (x, y(x), Dy(x)) dσ(x),

Occupation measure

Boundary measure

x

y(x)

µ(A × B)

A × B

Occupation and boundary measures

Milan Korda 12

µ(A × B × C) =

• Ω

IA×B×C (x, y(x), Dy(x)) dx

µ∂(A × B × C) =

• ∂Ω

IA×B×C (x, y(x), Dy(x)) dσ(x),

Occupation measure

Boundary measure

(µ, µ∂) ← → y(·)

Occupation and boundary measures

Milan Korda 13

Integration along solutions to the PDE → spatial integration w.r.t. µ and µ∂

• Ω

h (x, y(x), Dy(x)) dx =

• Ω×Y×Z

h (x, y, z) dµ(x, y, z)

for all h ∈ L∞

Occupation and boundary measures

Milan Korda 14

Integration along solutions to the PDE → spatial integration w.r.t. µ and µ∂

• Ω

h (x, y(x), Dy(x)) dx =

• Ω×Y×Z

h (x, y, z) dµ(x, y, z)

• ∂Ω

h (x, y(x), Dy(x)) σ(x) =

• ∂Ω×Y×Z

h (x, y, z) dµ∂(x, y, z)

for all h ∈ L∞

for all h ∈ L∞

Occupation and boundary measures

Milan Korda 15

Integration along solutions to the PDE → spatial integration w.r.t. µ and µ∂

• Ω

h (x, y(x), Dy(x)) dx =

• Ω×Y×Z

h (x, y, z) dµ(x, y, z)

• ∂Ω

h (x, y(x), Dy(x)) σ(x) =

• ∂Ω×Y×Z

h (x, y, z) dµ∂(x, y, z)

⇒

for all h ∈ L∞

for all h ∈ L∞

= R

Ω L (x, y, z) dµ(x, y, z) +

R

∂Ω L∂ (x, y, z) dµ∂(x, y, z)

J

• y(·)
• =

R

Ω L (x, y(x), Dy(x)) dx +

R

∂Ω L∂ (x, y(x), Dy(x)) dσ(x)

Milan Korda 16

When do µ and µ∂ come from our PDE?

Milan Korda 17

When do µ and µ∂ come from our PDE?

• ∂Ω×Y×Z

φ(x, y)G(x, y, z) dµ∂(x, y, z) = 0

• Ω×Y×Z

φ(x, y)F(x, y, z) dµ(x, y, z) = 0

• ∂Ω×Y×Z

φ(x, y)η(x) dµ∂(x, y, z) −

• Ω×Y×Z

∂φ ∂x + ∂φ ∂y z dµ(x, y, z) = 0.

• ∂Ω×Y×Z

ψ(x) dµ∂(x, y, z) =

• ∂Ω

ψ(x) σ(x)

for all φ ∈ C∞(Ω × Y) and ψ ∈ C(Ω)

Milan Korda 18

When do µ and µ∂ come from our PDE?

• ∂Ω×Y×Z

φ(x, y)G(x, y, z) dµ∂(x, y, z) = 0

• Ω×Y×Z

φ(x, y)F(x, y, z) dµ(x, y, z) = 0

• ∂Ω×Y×Z

φ(x, y)η(x) dµ∂(x, y, z) −

• Ω×Y×Z

∂φ ∂x + ∂φ ∂y z dµ(x, y, z) = 0.

System of linear equations in µ,µ∂

• ∂Ω×Y×Z

ψ(x) dµ∂(x, y, z) =

• ∂Ω

ψ(x) σ(x)

for all φ ∈ C∞(Ω × Y) and ψ ∈ C(Ω)

Infinite-dimensional LP

Milan Korda 19

inf / sup

y(·)

J

• y(·)
• subject to

y(·) solves the PDE

• Ω×Y×Z

φ(x, y)F(x, y, z) dµ(x, y, z) = 0 ∀φ ∈ C(Ω × Y )

• ∂Ω×Y×Z

φ(x, y)η(x) dµ∂(x, y, z) −

• Ω×Y×Z

∂φ ∂x + ∂φ ∂y z dµ(x, y, z) = 0 ∀φ ∈ C(Ω × Y )

• ∂Ω×Y×Z

φ(x, y)G(x, y, z) dµ∂(x, y, z) = 0 ∀φ ∈ C(Ω × Y ) ∀φ ∈ C(Ω × Y )

• ∂Ω×Y×Z

ψ(x) dµ∂(x, y, z) =

• ∂Ω

ψ(x) σ(x) ∀ψ ∈ C(Ω)

inf / sup

µ,µ∂

• Ω

L (x, y, z) dµ(x, y, z) +

• ∂Ω

L∂ (x, y, z) dµ∂(x, y, z)

µ ∈ M(Ω × Y × Z)+, µ∂ ∈ M(∂Ω × Y × Z)+

Infinite-dimensional LP

Milan Korda 20

A(µ, µ∂) = b

(µ, µ∂) ∈ K

inf / sup

µ,µ∂

• (µ, µ∂), c
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s.t.

inf / sup

y(·)

J

• y(·)
• subject to

y(·) solves the PDE

Infinite-dimensional LP

Milan Korda 21

A(µ, µ∂) = b

(µ, µ∂) ∈ K

inf / sup

µ,µ∂

• (µ, µ∂), c
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s.t.

Lemma: pLP is a lower/upper bound on p

inf / sup

y(·)

J

• y(·)
• subject to

y(·) solves the PDE

• ptimal value p

• ptimal value pLP

Relaxation gap

Milan Korda 22

When do we have p = pLP?

→ additional constraints added to the LP - “entropy inequalities”

• Proven for ODEs

• Proven for scalar hyperbolic conservation laws in [Marx et. al, 2018]

• General case open

Finite-dimensional approximation

Milan Korda 23

Optimization

• ver measures

Optimization over moment sequences Optimization over moment sequences up to degree d

SDP

⇒

⇔

Compactness

Finite-dimensional approximation

Milan Korda 24

Optimization

• ver measures

Optimization over moment sequences Optimization over moment sequences up to degree d

SDP

⇒

⇔

SDP solver

(MOSEK, SeDuMi,…)

High-level modelling

(Gloptipoly 3)

Finite-dimensional approximation

Milan Korda 25

Optimization

• ver measures

Optimization over moment sequences Optimization over moment sequences up to degree d

SDP

⇒

⇔

Theorem: pk ≤ pLP ≤ p and limk→∞ pk = pLP

Numerical examples – Burgers’ equation

Milan Korda 26

∂y ∂x1 + y ∂y ∂x2 = 0

Ω = [0, 5] × [0, 1]

• y(0, x2) = 10(x2(1 − x2))2

• y(x1, 0) = y(x1, 1)

Numerical examples – Burgers’ equation

Milan Korda 27

∂y ∂x1 + y ∂y ∂x2 = 0

Ω = [0, 5] × [0, 1]

• y(0, x2) = 10(x2(1 − x2))2

• y(x1, 0) = y(x1, 1)

Case 1: J(y(·)) = R 5 R 1

0 y(x1, x2)2 dx2dx1

Numerical examples – Burgers’ equation

Milan Korda 28

∂y ∂x1 + y ∂y ∂x2 = 0

Ω = [0, 5] × [0, 1]

• y(0, x2) = 10(x2(1 − x2))2

• y(x1, 0) = y(x1, 1)

Case 1: J(y(·)) = R 5 R 1

0 y(x1, x2)2 dx2dx1

Analytical solution: J(y(·)) = 50

63 ≈ 0.79365079365

pLB ≈ 0.79365079357

pUB ≈ 0.79365080188

SDP bounds:

(d = 4)

Numerical examples – Burgers’ equation

Milan Korda 29

∂y ∂x1 + y ∂y ∂x2 = 0

Ω = [0, 5] × [0, 1]

• y(0, x2) = 10(x2(1 − x2))2

• y(x1, 0) = y(x1, 1)

Case 2: J(y(·)) = R 5 R 1

0 x2 2y(x1, x2)2 dx2dx1

Numerical examples – Burgers’ equation

Milan Korda 30

∂y ∂x1 + y ∂y ∂x2 = 0

Ω = [0, 5] × [0, 1]

• y(0, x2) = 10(x2(1 − x2))2

• y(x1, 0) = y(x1, 1)

d 4 6 8 Lower bound (SeDuMi) 0.206 0.263 0.276 Upper bound (SeDuMi) 0.380 0.297 0.283 Parse time (Gloptipoly 3) 2.91s 3.41 s 6.23 s SDP solve time (SeDuMi / MOSEK) 2.62 / 1.63 s 2.61 / 1.32 s 20.67 / 7.05 s

Case 2: J(y(·)) = R 5 R 1

0 x2 2y(x1, x2)2 dx2dx1

Control – problem setup

Milan Korda 31

F(x, y(x), Dy(x)) = C(x, y(x))u(x) for x ∈ Ωo G(x, y(x), Dy(x)) = C∂(x, y(x))u∂(x) for x ∈ ∂Ω,

Control – problem setup

Milan Korda 32

F(x, y(x), Dy(x)) = C(x, y(x))u(x) for x ∈ Ωo G(x, y(x), Dy(x)) = C∂(x, y(x))u∂(x) for x ∈ ∂Ω,

+ R

Ω Lu (x, y(x)) u(x) dx +

R

∂Ω Lu∂ (x, y(x)) u dσ(x)

J

• y(·)
• :=

R

Ω L (x, y(x), Dy(x)) dx +

R

∂Ω L∂ (x, y(x), Dy(x)) dσ(x)

Goal: solve (at least approximately)

inf

y,u,u∂

J

• y, u, u∂
• subject to

y solves the PDE(u, u∂)

0 ≤ u ≤ 1 on Ω

0 ≤ u∂ ≤ 1 on ∂Ω

Infinite dimensional linear programming

Milan Korda 33

Aµ = b

µ ∈ M+

⇒

min

µ hg, µi

Infinite-dimensional LP

Nonlinear PDE

ANy = bN

min

y hgN, yi

y ∈ M+

N

⇒

Convex semidefinite program

with control

Control measures

Milan Korda 34

ν(A × B) :=

• Ω

IA×B (x, y(x)) u(x)dx

ν ⌧ ¯ µ with density u

⇔

Constraints easy to impose:

ν ≤ ¯ µ ⇔ u ∈ [0, 1]

Control measures

Milan Korda 35

ν∂(A × B) :=

• ∂Ω

IA×B (x, y(x)) u∂(x)dx

ν(A × B) :=

• Ω

IA×B (x, y(x)) u(x)dx

ν∂ ⌧ ¯ µ∂ with density u

ν ⌧ ¯ µ with density u

⇔

Constraints easy to impose:

ν ≤ ¯ µ ⇔ u ∈ [0, 1]

ν∂ ≤ ¯ µ∂ ⇔ u∂ ∈ [0, 1]

Infinite-dimensional LP – with control

Milan Korda 36

s.t.

inf

y,u,u∂

J

• y, u, u∂
• subject to

y solves the PDE(u, u∂)

0 ≤ u ≤ 1 on Ω

0 ≤ u∂ ≤ 1 on ∂Ω

inf / sup

µ,µ∂,ν,ν∂

• (µ, µ∂, ν, ν∂), c
• <latexit sha1_base64="z8+43EctzPQ7D8MyPzeMcTqEZQE=">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</latexit><latexit sha1_base64="+az6SGfNpftGVwD8s6d4WE70/gE=">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</latexit><latexit sha1_base64="+az6SGfNpftGVwD8s6d4WE70/gE=">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</latexit>

A(µ, µ∂, ν, ν∂) = b

(µ, µ∂, ν, ν∂) ∈ K

Infinite-dimensional LP – with control

Milan Korda 37

• ptimal value p

• ptimal value pLP

inf

y,u,u∂

J

• y, u, u∂
• subject to

y solves the PDE(u, u∂)

0 ≤ u ≤ 1 on Ω

0 ≤ u∂ ≤ 1 on ∂Ω

Lemma: pLP is a lower bound on p

s.t.

inf / sup

µ,µ∂,ν,ν∂

• (µ, µ∂, ν, ν∂), c
• <latexit sha1_base64="z8+43EctzPQ7D8MyPzeMcTqEZQE=">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</latexit><latexit sha1_base64="+az6SGfNpftGVwD8s6d4WE70/gE=">AGqHicjZTtbiM1FIYn3RK6gYUWfi4SI9pdFUhLsgIhCpVsNsW7RbtVk1bVIfK9pxMrPpjZHvaFMv3xi3wk7vBzqRtZjaVcDTSid/nvGfsMzYpODO21/u3tfRo+YP2hyuPOx9/OST1fXPjsxqtQUBlRxpc8INsCZhIFlsNZoQELwuGUXP4a9dMr0IYpeWxvChgKnEs2YhTbMHWx+jcqZRZ0sA4FN+2QKLvhuUAF1pZh7rtxPndIlnPhvexDBlETx+TIo+53qFv9NWXhfdpJ0c/xlyLCcsSxzDmkm3eFKsNp+L/rfd2lZueunUuVtd7273pSN8P+rNgfeL5Mty9XDw9mJt+R+UKVoKkJZybMx5v1fYoYv+lIPvoNJAgeklzuE8hBILMEM3WqfPgszWTpSOjzSptPZ+QyHhTE3gRSYDs2TS1OLtSIqFV2VBDN8rGtv89xf+hiYZC0jluYWKpEUcezK1aY2QIm1Qo6tbToZQpomE1KyajKoDHL7cRqHB1itWuW2XG6k/7Q6wlRd8U8V5rZsWA0hmMcMltrDUCbEsNxh3iomAy3wlOW9EtlArfoFYCy8idN/WhOwB+BaFOP0dSkjfzDYnZBkszTpF8WzvRDs3MOR3TqEjJXCdztpJH6zmDPa4OZNH6r1LBYTSqpYzIUAU9Vo2u0WS7gO3QiLyRx6FY9I6LcWDpFR+sr7uh6SiD8PjZ2GSjvCS/Dpen8BqGughuwBLq9xeypsud3XAPIBXtT4w7AWafEiVlRvcLugBQC5BwhZABTAS0Ip4UxRdAR8TfWRw1xd0JlLM3eumejCvHjTVvXl1r6nuz6v7TfXlvPrS1/v8DWLhWGqmdFzdev/PCjUjp3xE64dqUuLw050oqBhvgrHJNzmFgTzbgNdUYi2Do3jrYoM1aywhv0FG4iUnIPd8H7Dh0uw37zy3g8GL7Z/2u69C5fh90k1VpKnyVfJZtJPfkx2k4PkbTJIaOt5601r0Dpf9t+1z5t/1GhS61ZzudJbTJf2pfZPo=</latexit><latexit sha1_base64="+az6SGfNpftGVwD8s6d4WE70/gE=">AGqHicjZTtbiM1FIYn3RK6gYUWfi4SI9pdFUhLsgIhCpVsNsW7RbtVk1bVIfK9pxMrPpjZHvaFMv3xi3wk7vBzqRtZjaVcDTSid/nvGfsMzYpODO21/u3tfRo+YP2hyuPOx9/OST1fXPjsxqtQUBlRxpc8INsCZhIFlsNZoQELwuGUXP4a9dMr0IYpeWxvChgKnEs2YhTbMHWx+jcqZRZ0sA4FN+2QKLvhuUAF1pZh7rtxPndIlnPhvexDBlETx+TIo+53qFv9NWXhfdpJ0c/xlyLCcsSxzDmkm3eFKsNp+L/rfd2lZueunUuVtd7273pSN8P+rNgfeL5Mty9XDw9mJt+R+UKVoKkJZybMx5v1fYoYv+lIPvoNJAgeklzuE8hBILMEM3WqfPgszWTpSOjzSptPZ+QyHhTE3gRSYDs2TS1OLtSIqFV2VBDN8rGtv89xf+hiYZC0jluYWKpEUcezK1aY2QIm1Qo6tbToZQpomE1KyajKoDHL7cRqHB1itWuW2XG6k/7Q6wlRd8U8V5rZsWA0hmMcMltrDUCbEsNxh3iomAy3wlOW9EtlArfoFYCy8idN/WhOwB+BaFOP0dSkjfzDYnZBkszTpF8WzvRDs3MOR3TqEjJXCdztpJH6zmDPa4OZNH6r1LBYTSqpYzIUAU9Vo2u0WS7gO3QiLyRx6FY9I6LcWDpFR+sr7uh6SiD8PjZ2GSjvCS/Dpen8BqGughuwBLq9xeypsud3XAPIBXtT4w7AWafEiVlRvcLugBQC5BwhZABTAS0Ip4UxRdAR8TfWRw1xd0JlLM3eumejCvHjTVvXl1r6nuz6v7TfXlvPrS1/v8DWLhWGqmdFzdev/PCjUjp3xE64dqUuLw050oqBhvgrHJNzmFgTzbgNdUYi2Do3jrYoM1aywhv0FG4iUnIPd8H7Dh0uw37zy3g8GL7Z/2u69C5fh90k1VpKnyVfJZtJPfkx2k4PkbTJIaOt5601r0Dpf9t+1z5t/1GhS61ZzudJbTJf2pfZPo=</latexit>

A(µ, µ∂, ν, ν∂) = b

(µ, µ∂, ν, ν∂) ∈ K

Controller extraction

Milan Korda 38

Optimization

• ver measures

Optimization over moment sequences Optimization over moment sequences up to degree d

SDP

⇒

⇔

u(x1, x2) = κ(y(x1, x2), x1, x2)

u∂(x1, x2) = κ∂(y(x1, x2), x1, x2)

Feedback controller

Numerical examples with control

Milan Korda 39

∂y ∂x1 + y ∂y ∂x2 = u(x1, x2)

• y(0, x2) = 10(x2(1 − x2))2

• y(x1, 0) = y(x1, 1)

Ω = [0, 3] × [0, 1]

Goal: minimize R 3 R 1

0 y(x1, x2)2dx2dx1

Numerical examples with control

Milan Korda 40

∂y ∂x1 + y ∂y ∂x2 = u(x1, x2)

• y(0, x2) = 10(x2(1 − x2))2

• y(x1, 0) = y(x1, 1)

Ω = [0, 3] × [0, 1]

Goal: minimize R 3 R 1

0 y(x1, x2)2dx2dx1

x1 (= time)

x2

y(x1, x2)

x2

x1 ( = t i m e )

u(x1, x2)

SDP → polynomial controller of degree 3

Computational complexity

Milan Korda 41

n = dim(x) + dim(y) + # derivatives appearing nonlinarly

Measures supported on subsets of Rn of dimension

n 4 6 8 10 12 17 21 24 d = 4 15 28 45 66 91 171 253 325 d = 6 35 84 165 286 455 1140 2024 2925 d = 8 70 210 495 1001 1820 5985 12650 20475

Largest SDP block of size: N = n+d/2

n

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Degree bounding

Complexity reduction: Sparsity

Conclusion

Milan Korda 42

• Convex optimization based approach to analysis and control of PDEs

→ Off-the-shelf software available

→ Discontinuities (e.g., shocks) well resolved

• No spatio-temporal gridding

• Solutions represented by measures supported on their graphs

[Marx et al, 2018]

Conclusion

Milan Korda 43

• Convex optimization based approach to analysis and control of PDEs

Future directions

• Absence of relaxation gap

• More extensive numerical experiments

→ Off-the-shelf software available

→ Discontinuities (e.g., shocks) well resolved

• No spatio-temporal gridding

• Solutions represented by measures supported on their graphs

Preprint:

[Marx et al, 2018]