![Page 1: L2 Stabilization of Coupled Viscous Burgers' Equations · 1 3 u3(1;t) Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations. 7/24 First approach:](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb50bf2389f441294151607/html5/thumbnails/1.jpg)
L2 Stabilization of Coupled Viscous Burgers’Equations
Saad Qadeer & Jean-Baptiste Sibille
UC Berkeley
May 8th, 2014
1/24
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Objective of the project
Goal: stabilize the system:
ut(x , t)− ε1uxx(x , t) + u(x , t)ux(x , t) = 0, x ∈ [0, 1]u(0, t) = 0
ux(1, t) = U(t)vt(x , t)− ε2vxx(x , t) + v(x , t)vx(x , t) = 0, x ∈ [1, 1+ D]
v(1, t) = qu(1, t)vx(1+ D, t) = W (t)
whereε1, ε2,D > 0, q ∈ R.U,W are the control inputs to the system.
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Motivation
Interconnected roads example
Figure: Real life example of the coupled Burgers’ equations
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4/24
First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
1 First approach: with only one equationProblem FormulationLyapunov Stability & Controller DesignSimulation
2 Back to the coupled problemLyapunov Stability & Controller design: coupled caseSimulation
3 Current/Future Work: the inviscid case
Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations
![Page 5: L2 Stabilization of Coupled Viscous Burgers' Equations · 1 3 u3(1;t) Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations. 7/24 First approach:](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb50bf2389f441294151607/html5/thumbnails/5.jpg)
5/24
First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
Problem FormulationLyapunov Stability & Controller DesignSimulation
Table of Contents
1 First approach: with only one equationProblem FormulationLyapunov Stability & Controller DesignSimulation
2 Back to the coupled problemLyapunov Stability & Controller design: coupled caseSimulation
3 Current/Future Work: the inviscid case
Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations
![Page 6: L2 Stabilization of Coupled Viscous Burgers' Equations · 1 3 u3(1;t) Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations. 7/24 First approach:](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb50bf2389f441294151607/html5/thumbnails/6.jpg)
6/24
First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
Problem FormulationLyapunov Stability & Controller DesignSimulation
Problem Formulation
We first consider the system
ut(x , t)− ε1uxx(x , t) + u(x , t)ux(x , t) = 0, x ∈ [0, 1]u(0, t) = 0
ux(1, t) = U(t)
Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations
![Page 7: L2 Stabilization of Coupled Viscous Burgers' Equations · 1 3 u3(1;t) Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations. 7/24 First approach:](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb50bf2389f441294151607/html5/thumbnails/7.jpg)
7/24
First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
Problem FormulationLyapunov Stability & Controller DesignSimulation
Lyapunov Function
Let’s define the Lyapunov function (L2 norm of u):
V (t) =12
∫ 1
0u(x , t)2dx
And take its derivative:
V̇ (t) = ε1U(t)u(1, t)− ε1∫ 1
0u2x (x , t)dx − 1
3u3(1, t)
Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations
![Page 8: L2 Stabilization of Coupled Viscous Burgers' Equations · 1 3 u3(1;t) Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations. 7/24 First approach:](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb50bf2389f441294151607/html5/thumbnails/8.jpg)
7/24
First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
Problem FormulationLyapunov Stability & Controller DesignSimulation
Lyapunov Function
Let’s define the Lyapunov function (L2 norm of u):
V (t) =12
∫ 1
0u(x , t)2dx
And take its derivative:
V̇ (t) = ε1U(t)u(1, t)− ε1∫ 1
0u2x (x , t)dx − 1
3u3(1, t)
Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations
![Page 9: L2 Stabilization of Coupled Viscous Burgers' Equations · 1 3 u3(1;t) Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations. 7/24 First approach:](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb50bf2389f441294151607/html5/thumbnails/9.jpg)
8/24
First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
Problem FormulationLyapunov Stability & Controller DesignSimulation
Controller Design
We choose a controller:
U(t) = −c(u(1, t) + u(1, t)3), with c ≥ 16ε1
which leads to:
V̇ (t) = −ε1∫ 1
0u2xdx − ε1c(u(1, t) + u3(1, t))u(1, t)− 1
3u3(1, t)
= −ε1∫ 1
0u2xdx − ε1c
[1+
u(1, t)3ε1c
+ u2(1, t)]
︸ ︷︷ ︸≥0 with this choice of c
u2(1, t)
≤ −ε1∫ 1
0u2xdx
Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations
![Page 10: L2 Stabilization of Coupled Viscous Burgers' Equations · 1 3 u3(1;t) Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations. 7/24 First approach:](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb50bf2389f441294151607/html5/thumbnails/10.jpg)
8/24
First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
Problem FormulationLyapunov Stability & Controller DesignSimulation
Controller Design
We choose a controller:
U(t) = −c(u(1, t) + u(1, t)3), with c ≥ 16ε1
which leads to:
V̇ (t) = −ε1∫ 1
0u2xdx − ε1c(u(1, t) + u3(1, t))u(1, t)− 1
3u3(1, t)
= −ε1∫ 1
0u2xdx − ε1c
[1+
u(1, t)3ε1c
+ u2(1, t)]
︸ ︷︷ ︸≥0 with this choice of c
u2(1, t)
≤ −ε1∫ 1
0u2xdx
Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations
![Page 11: L2 Stabilization of Coupled Viscous Burgers' Equations · 1 3 u3(1;t) Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations. 7/24 First approach:](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb50bf2389f441294151607/html5/thumbnails/11.jpg)
8/24
First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
Problem FormulationLyapunov Stability & Controller DesignSimulation
Controller Design
We choose a controller:
U(t) = −c(u(1, t) + u(1, t)3), with c ≥ 16ε1
which leads to:
V̇ (t) = −ε1∫ 1
0u2xdx − ε1c(u(1, t) + u3(1, t))u(1, t)− 1
3u3(1, t)
= −ε1∫ 1
0u2xdx − ε1c
[1+
u(1, t)3ε1c
+ u2(1, t)]
︸ ︷︷ ︸≥0 with this choice of c
u2(1, t)
≤ −ε1∫ 1
0u2xdx
Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations
![Page 12: L2 Stabilization of Coupled Viscous Burgers' Equations · 1 3 u3(1;t) Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations. 7/24 First approach:](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb50bf2389f441294151607/html5/thumbnails/12.jpg)
8/24
First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
Problem FormulationLyapunov Stability & Controller DesignSimulation
Controller Design
We choose a controller:
U(t) = −c(u(1, t) + u(1, t)3), with c ≥ 16ε1
which leads to:
V̇ (t) = −ε1∫ 1
0u2xdx − ε1c(u(1, t) + u3(1, t))u(1, t)− 1
3u3(1, t)
= −ε1∫ 1
0u2xdx − ε1c
[1+
u(1, t)3ε1c
+ u2(1, t)]
︸ ︷︷ ︸≥0 with this choice of c
u2(1, t)
≤ −ε1∫ 1
0u2xdx
Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations
![Page 13: L2 Stabilization of Coupled Viscous Burgers' Equations · 1 3 u3(1;t) Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations. 7/24 First approach:](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb50bf2389f441294151607/html5/thumbnails/13.jpg)
9/24
First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
Problem FormulationLyapunov Stability & Controller DesignSimulation
V̇ (t) ≤ −ε1∫ 1
0u2xdx
≤ −ε1∫ 1
0u2(x , t)dx ∵ Poincaré’s inequality
which leads to
V̇ (t) ≤ −2ε1V (t)
Hence
Exponential Stability
V (t) ≤ V (0) exp(−2ε1t)
Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations
![Page 14: L2 Stabilization of Coupled Viscous Burgers' Equations · 1 3 u3(1;t) Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations. 7/24 First approach:](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb50bf2389f441294151607/html5/thumbnails/14.jpg)
9/24
First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
Problem FormulationLyapunov Stability & Controller DesignSimulation
V̇ (t) ≤ −ε1∫ 1
0u2xdx
≤ −ε1∫ 1
0u2(x , t)dx ∵ Poincaré’s inequality
which leads to
V̇ (t) ≤ −2ε1V (t)
Hence
Exponential Stability
V (t) ≤ V (0) exp(−2ε1t)
Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations
![Page 15: L2 Stabilization of Coupled Viscous Burgers' Equations · 1 3 u3(1;t) Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations. 7/24 First approach:](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb50bf2389f441294151607/html5/thumbnails/15.jpg)
9/24
First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
Problem FormulationLyapunov Stability & Controller DesignSimulation
V̇ (t) ≤ −ε1∫ 1
0u2xdx
≤ −ε1∫ 1
0u2(x , t)dx ∵ Poincaré’s inequality
which leads to
V̇ (t) ≤ −2ε1V (t)
Hence
Exponential Stability
V (t) ≤ V (0) exp(−2ε1t)
Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations
![Page 16: L2 Stabilization of Coupled Viscous Burgers' Equations · 1 3 u3(1;t) Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations. 7/24 First approach:](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb50bf2389f441294151607/html5/thumbnails/16.jpg)
9/24
First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
Problem FormulationLyapunov Stability & Controller DesignSimulation
V̇ (t) ≤ −ε1∫ 1
0u2xdx
≤ −ε1∫ 1
0u2(x , t)dx ∵ Poincaré’s inequality
which leads to
V̇ (t) ≤ −2ε1V (t)
Hence
Exponential Stability
V (t) ≤ V (0) exp(−2ε1t)
Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations
![Page 17: L2 Stabilization of Coupled Viscous Burgers' Equations · 1 3 u3(1;t) Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations. 7/24 First approach:](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb50bf2389f441294151607/html5/thumbnails/17.jpg)
10/24
First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
Problem FormulationLyapunov Stability & Controller DesignSimulation
Figure: Simulation using finite differences with ∆x = 0.1,∆t = 0.005
Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations
![Page 18: L2 Stabilization of Coupled Viscous Burgers' Equations · 1 3 u3(1;t) Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations. 7/24 First approach:](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb50bf2389f441294151607/html5/thumbnails/18.jpg)
11/24
First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
Lyapunov Stability & Controller design: coupled caseSimulation
Table of Contents
1 First approach: with only one equationProblem FormulationLyapunov Stability & Controller DesignSimulation
2 Back to the coupled problemLyapunov Stability & Controller design: coupled caseSimulation
3 Current/Future Work: the inviscid case
Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations
![Page 19: L2 Stabilization of Coupled Viscous Burgers' Equations · 1 3 u3(1;t) Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations. 7/24 First approach:](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb50bf2389f441294151607/html5/thumbnails/19.jpg)
12/24
First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
Lyapunov Stability & Controller design: coupled caseSimulation
Recall the system
Goal: stabilize the system:
ut(x , t)− ε1uxx(x , t) + u(x , t)ux(x , t) = 0, x ∈ [0, 1]u(0, t) = 0
ux(1, t) = U(t)vt(x , t)− ε2vxx(x , t) + v(x , t)vx(x , t) = 0, x ∈ [1, 1+ D]
v(1, t) = qu(1, t)vx(1+ D, t) = W (t)
whereε1, ε2,D > 0, q ∈ R.U,W are the control inputs to the system.
Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations
![Page 20: L2 Stabilization of Coupled Viscous Burgers' Equations · 1 3 u3(1;t) Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations. 7/24 First approach:](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb50bf2389f441294151607/html5/thumbnails/20.jpg)
13/24
First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
Lyapunov Stability & Controller design: coupled caseSimulation
Lyapunov Function
Let’s define the Lyapunov function
V (t) =12
∫ 1
0u(x , t)2dx +
12
∫ 1+D
1v(x , t)2dx
We have:
V̇ (t) = −ε1∫ 1
0ux(x , t)2dx − ε2
∫ 1+D
1vx(x , t)2dx
+ u(1, t)(ε1U(t) +
13(q3 − 1)u(1, t)2 − ε2qvx(1, t)
)+ v(1+ D, t)
(ε2W (t)− 1
3v(1+ D, t)2
)
Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations
![Page 21: L2 Stabilization of Coupled Viscous Burgers' Equations · 1 3 u3(1;t) Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations. 7/24 First approach:](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb50bf2389f441294151607/html5/thumbnails/21.jpg)
13/24
First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
Lyapunov Stability & Controller design: coupled caseSimulation
Lyapunov Function
Let’s define the Lyapunov function
V (t) =12
∫ 1
0u(x , t)2dx +
12
∫ 1+D
1v(x , t)2dx
We have:
V̇ (t) = −ε1∫ 1
0ux(x , t)2dx − ε2
∫ 1+D
1vx(x , t)2dx
+ u(1, t)(ε1U(t) +
13(q3 − 1)u(1, t)2 − ε2qvx(1, t)
)+ v(1+ D, t)
(ε2W (t)− 1
3v(1+ D, t)2
)
Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations
![Page 22: L2 Stabilization of Coupled Viscous Burgers' Equations · 1 3 u3(1;t) Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations. 7/24 First approach:](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb50bf2389f441294151607/html5/thumbnails/22.jpg)
14/24
First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
Lyapunov Stability & Controller design: coupled caseSimulation
Controller Design
We choose the control laws as
U(t) =ε2ε1
qvx(1, t)−13ε1
(q3 − 1)u(1, t)2
W (t) =13ε2
v(1+ D, t)2 − 2D
v(1+ D, t)
We get
V̇ (t) = −ε1∫ 1
0ux(x , t)2dx − ε2
∫ 1+D
1vx(x , t)2dx
+ u(1, t)(ε1U(t) +
13(q3 − 1)u(1, t)2 − ε2qvx(1, t)
)+ v(1+ D, t)
(ε2W (t)− 1
3v(1+ D, t)2
)
Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations
![Page 23: L2 Stabilization of Coupled Viscous Burgers' Equations · 1 3 u3(1;t) Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations. 7/24 First approach:](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb50bf2389f441294151607/html5/thumbnails/23.jpg)
14/24
First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
Lyapunov Stability & Controller design: coupled caseSimulation
Controller Design
We choose the control laws as
U(t) =ε2ε1
qvx(1, t)−13ε1
(q3 − 1)u(1, t)2
W (t) =13ε2
v(1+ D, t)2 − 2D
v(1+ D, t)
We get
V̇ (t) = −ε1∫ 1
0ux(x , t)2dx − ε2
∫ 1+D
1vx(x , t)2dx
+ u(1, t)(ε1U(t) +
13(q3 − 1)u(1, t)2 − ε2qvx(1, t)
)+ v(1+ D, t)
(ε2W (t)− 1
3v(1+ D, t)2
)
Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations
![Page 24: L2 Stabilization of Coupled Viscous Burgers' Equations · 1 3 u3(1;t) Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations. 7/24 First approach:](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb50bf2389f441294151607/html5/thumbnails/24.jpg)
14/24
First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
Lyapunov Stability & Controller design: coupled caseSimulation
Controller Design
We choose the control laws as
U(t) =ε2ε1
qvx(1, t)−13ε1
(q3 − 1)u(1, t)2
W (t) =13ε2
v(1+ D, t)2 − 2D
v(1+ D, t)
We get
V̇ (t) = −ε1∫ 1
0ux(x , t)2dx − ε2
∫ 1+D
1vx(x , t)2dx
+ u(1, t)(ε1U(t) +
13(q3 − 1)u(1, t)2 − ε2qvx(1, t)
)+ v(1+ D, t)
(ε2W (t)− 1
3v(1+ D, t)2
)Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations
![Page 25: L2 Stabilization of Coupled Viscous Burgers' Equations · 1 3 u3(1;t) Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations. 7/24 First approach:](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb50bf2389f441294151607/html5/thumbnails/25.jpg)
15/24
First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
Lyapunov Stability & Controller design: coupled caseSimulation
V̇ (t) = −ε1∫ 1
0ux(x , t)2dx − ε2
∫ 1+D
1vx(x , t)2dx
−2ε2D
v(1+ D, t)2
≤ −ε1∫ 1
0u(x , t)2dx − ε2
D2
∫ 1+D
1v(x , t)2dx
In the case D ≤ 1, we use −1D2 ≤ −1 to get
V̇ (t) ≤ −ε1∫ 1
0u(x , t)2dx − ε2
∫ 1+D
1v(x , t)2dx
≤ −2min{ε1, ε2}(12
∫ 1
0u(x , t)2dx +
12
∫ 1+D
1v(x , t)2dx
)= −2min{ε1, ε2}V (t)
Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations
![Page 26: L2 Stabilization of Coupled Viscous Burgers' Equations · 1 3 u3(1;t) Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations. 7/24 First approach:](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb50bf2389f441294151607/html5/thumbnails/26.jpg)
15/24
First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
Lyapunov Stability & Controller design: coupled caseSimulation
V̇ (t) = −ε1∫ 1
0ux(x , t)2dx − ε2
∫ 1+D
1vx(x , t)2dx
−2ε2D
v(1+ D, t)2
≤ −ε1∫ 1
0u(x , t)2dx − ε2
D2
∫ 1+D
1v(x , t)2dx
In the case D ≤ 1, we use −1D2 ≤ −1 to get
V̇ (t) ≤ −ε1∫ 1
0u(x , t)2dx − ε2
∫ 1+D
1v(x , t)2dx
≤ −2min{ε1, ε2}(12
∫ 1
0u(x , t)2dx +
12
∫ 1+D
1v(x , t)2dx
)= −2min{ε1, ε2}V (t)
Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations
![Page 27: L2 Stabilization of Coupled Viscous Burgers' Equations · 1 3 u3(1;t) Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations. 7/24 First approach:](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb50bf2389f441294151607/html5/thumbnails/27.jpg)
15/24
First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
Lyapunov Stability & Controller design: coupled caseSimulation
V̇ (t) = −ε1∫ 1
0ux(x , t)2dx − ε2
∫ 1+D
1vx(x , t)2dx
−2ε2D
v(1+ D, t)2
≤ −ε1∫ 1
0u(x , t)2dx − ε2
D2
∫ 1+D
1v(x , t)2dx
In the case D ≤ 1, we use −1D2 ≤ −1 to get
V̇ (t) ≤ −ε1∫ 1
0u(x , t)2dx − ε2
∫ 1+D
1v(x , t)2dx
≤ −2min{ε1, ε2}(12
∫ 1
0u(x , t)2dx +
12
∫ 1+D
1v(x , t)2dx
)= −2min{ε1, ε2}V (t)
Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations
![Page 28: L2 Stabilization of Coupled Viscous Burgers' Equations · 1 3 u3(1;t) Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations. 7/24 First approach:](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb50bf2389f441294151607/html5/thumbnails/28.jpg)
15/24
First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
Lyapunov Stability & Controller design: coupled caseSimulation
V̇ (t) = −ε1∫ 1
0ux(x , t)2dx − ε2
∫ 1+D
1vx(x , t)2dx
−2ε2D
v(1+ D, t)2
≤ −ε1∫ 1
0u(x , t)2dx − ε2
D2
∫ 1+D
1v(x , t)2dx
In the case D ≤ 1, we use −1D2 ≤ −1 to get
V̇ (t) ≤ −ε1∫ 1
0u(x , t)2dx − ε2
∫ 1+D
1v(x , t)2dx
≤ −2min{ε1, ε2}(12
∫ 1
0u(x , t)2dx +
12
∫ 1+D
1v(x , t)2dx
)
= −2min{ε1, ε2}V (t)
Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations
![Page 29: L2 Stabilization of Coupled Viscous Burgers' Equations · 1 3 u3(1;t) Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations. 7/24 First approach:](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb50bf2389f441294151607/html5/thumbnails/29.jpg)
15/24
First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
Lyapunov Stability & Controller design: coupled caseSimulation
V̇ (t) = −ε1∫ 1
0ux(x , t)2dx − ε2
∫ 1+D
1vx(x , t)2dx
−2ε2D
v(1+ D, t)2
≤ −ε1∫ 1
0u(x , t)2dx − ε2
D2
∫ 1+D
1v(x , t)2dx
In the case D ≤ 1, we use −1D2 ≤ −1 to get
V̇ (t) ≤ −ε1∫ 1
0u(x , t)2dx − ε2
∫ 1+D
1v(x , t)2dx
≤ −2min{ε1, ε2}(12
∫ 1
0u(x , t)2dx +
12
∫ 1+D
1v(x , t)2dx
)= −2min{ε1, ε2}V (t)
Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations
![Page 30: L2 Stabilization of Coupled Viscous Burgers' Equations · 1 3 u3(1;t) Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations. 7/24 First approach:](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb50bf2389f441294151607/html5/thumbnails/30.jpg)
16/24
First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
Lyapunov Stability & Controller design: coupled caseSimulation
In the case D ≥ 1, we use −1D2 ≥ −1 to get
V̇ (t) ≤ − ε1D2
∫ 1
0u(x , t)2dx − ε2
D2
∫ 1+D
1v(x , t)2dx
≤ − 2D2 min{ε1, ε2}
(12
∫ 1
0u(x , t)2dx +
12
∫ 1+D
1v(x , t)2dx
)= − 2
D2 min{ε1, ε2}V (t)
In the end:
V̇ (t) ≤ −2 min{ε1, ε2}max{1,D2}
V (t)
Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations
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16/24
First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
Lyapunov Stability & Controller design: coupled caseSimulation
In the case D ≥ 1, we use −1D2 ≥ −1 to get
V̇ (t) ≤ − ε1D2
∫ 1
0u(x , t)2dx − ε2
D2
∫ 1+D
1v(x , t)2dx
≤ − 2D2 min{ε1, ε2}
(12
∫ 1
0u(x , t)2dx +
12
∫ 1+D
1v(x , t)2dx
)
= − 2D2 min{ε1, ε2}V (t)
In the end:
V̇ (t) ≤ −2 min{ε1, ε2}max{1,D2}
V (t)
Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations
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16/24
First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
Lyapunov Stability & Controller design: coupled caseSimulation
In the case D ≥ 1, we use −1D2 ≥ −1 to get
V̇ (t) ≤ − ε1D2
∫ 1
0u(x , t)2dx − ε2
D2
∫ 1+D
1v(x , t)2dx
≤ − 2D2 min{ε1, ε2}
(12
∫ 1
0u(x , t)2dx +
12
∫ 1+D
1v(x , t)2dx
)= − 2
D2 min{ε1, ε2}V (t)
In the end:
V̇ (t) ≤ −2 min{ε1, ε2}max{1,D2}
V (t)
Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations
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16/24
First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
Lyapunov Stability & Controller design: coupled caseSimulation
In the case D ≥ 1, we use −1D2 ≥ −1 to get
V̇ (t) ≤ − ε1D2
∫ 1
0u(x , t)2dx − ε2
D2
∫ 1+D
1v(x , t)2dx
≤ − 2D2 min{ε1, ε2}
(12
∫ 1
0u(x , t)2dx +
12
∫ 1+D
1v(x , t)2dx
)= − 2
D2 min{ε1, ε2}V (t)
In the end:
V̇ (t) ≤ −2 min{ε1, ε2}max{1,D2}
V (t)
Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations
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17/24
First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
Lyapunov Stability & Controller design: coupled caseSimulation
Hence
Exponential Stability
V (t) ≤ V (0) exp(−2 min{ε1, ε2}
max{1,D2}t)
Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations
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18/24
First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
Lyapunov Stability & Controller design: coupled caseSimulation
Figure: Simulation using finite differences with ∆x = 0.1,∆t = 0.005. Thered lines represents x = 1.
Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations
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19/24
First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
Lyapunov Stability & Controller design: coupled caseSimulation
Figure: Plot of log(V (t)) against t and the best fit line.Theoretical slope ≤ −0.25. Simulated = −0.78.
Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations
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20/24
First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
Table of Contents
1 First approach: with only one equationProblem FormulationLyapunov Stability & Controller DesignSimulation
2 Back to the coupled problemLyapunov Stability & Controller design: coupled caseSimulation
3 Current/Future Work: the inviscid case
Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations
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21/24
First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
Problem formulation
ut(x , t) + u(x , t)ux(x , t) = 0, x ∈ (0, 1)u(0, t) = U0(t)u(1, t) = U1(t)
Issue: Even for smooth boundary conditions, the solution doesn’talways exist in a classical sense, but rather in a weak sense, withweak boundary conditions. Hence the solution can exhibit shocks.
Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations
![Page 39: L2 Stabilization of Coupled Viscous Burgers' Equations · 1 3 u3(1;t) Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations. 7/24 First approach:](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb50bf2389f441294151607/html5/thumbnails/39.jpg)
21/24
First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
Problem formulation
ut(x , t) + u(x , t)ux(x , t) = 0, x ∈ (0, 1)u(0, t) = U0(t)u(1, t) = U1(t)
Issue: Even for smooth boundary conditions, the solution doesn’talways exist in a classical sense, but rather in a weak sense, withweak boundary conditions. Hence the solution can exhibit shocks.
Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations
![Page 40: L2 Stabilization of Coupled Viscous Burgers' Equations · 1 3 u3(1;t) Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations. 7/24 First approach:](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb50bf2389f441294151607/html5/thumbnails/40.jpg)
22/24
First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
Equilibrium
Most general form of equilibrium for this equation:
Figure: Inviscid case equilibrium.
Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations
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First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
Conclusion
We achieved stabilization of the coupled Burgers’ equation.Not shown in this presentation:
Equilibria analysis for the viscous case.Constant control in the viscous case.Linearized form of the Burgers’ equation (heat equation).
Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations
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First approach: with only one equationBack to the coupled problem
Current/Future Work: the inviscid case
Questions?
Figure: McDonald’s approves Burgers’ equations
Saad Qadeer & Jean-Baptiste Sibille L2 Stabilization of Coupled Viscous Burgers’ Equations