local exponential stabilization of a 2 x 2 quasilinear ...fa/cdps/talks/vazquez.pdflocal exponential...
TRANSCRIPT
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Local Exponential Stabilizationof a 2 x 2 Quasilinear Hyperbolic
System using Backstepping
Rafael Vazquez, Miroslav Krstic, Jean-Michel Coron and Georges Bastin
CDPS, Wuppertal, 2011
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Outline
• 2x2 hyperbolic quasi-linear PDEs
• Backstepping control of the linearized system
• Well-posedness of kernel PDEs
• Stability of the nonlinear system
• Conclusions
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2x2 Hyperbolic Quasi-Linear PDEs
zt +Λ(z,x)zx+ f (z,x) = 0,
x∈ [0,1], where z : [0,1]× [0,∞)→R2, Λ : R2× [0,1]→M 2,2(R), f : R2× [0,1]→R
2.
Consider w.l.o.
Λ(0,x) =[
Λ1(x) 00 Λ2(x)
]
Λ1(x) and Λ2(x) are the speeds of propagation of z= [z1 z2]T . According to their signs:
homo directional hetero directional∀x∈ [0,1], Λ1(x)Λ2(x)>>> 0 ∀x∈ [0,1], Λ1(x)Λ2(x)<<< 0
Heterodirectional −→ one boundary condition on each side.
Homodirectional −→ two boundary conditions on the same side.
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2x2 Hyperbolic Quasi-Linear PDEs
zt +Λ(z,x)zx+ f (z,x) = 0,
x∈ [0,1], where z : [0,1]× [0,∞)→R2, Λ : R2× [0,1]→M 2,2(R), f : R2× [0,1]→R
2.
Consider w.l.o.
Λ(0,x) =[
Λ1(x) 00 Λ2(x)
]
Λ1(x) and Λ2(x) are the speeds of propagation of z= [z1 z2]T . According to their signs:
homodirectional heterodirectional∀x∈ [0,1], Λ1(x)Λ2(x)>>> 0 ∀x∈ [0,1], Λ1(x)Λ2(x)<<< 0
Heterodirectional −→ one boundary condition on each side.
Homodirectional −→ two boundary conditions on the same side.
![Page 5: Local Exponential Stabilization of a 2 x 2 Quasilinear ...fa/cdps/talks/Vazquez.pdfLocal Exponential Stabilization of a 2 x 2 Quasilinear Hyperbolic System using Backstepping Rafael](https://reader030.vdocument.in/reader030/viewer/2022040703/5dd0bf73d6be591ccb62807f/html5/thumbnails/5.jpg)
2x2 Hyperbolic Quasi-Linear PDEs
zt +Λ(z,x)zx+ f (z,x) = 0,
x∈ [0,1], where z : [0,1]× [0,∞)→R2, Λ : R2× [0,1]→M 2,2(R), f : R2× [0,1]→R
2.
Consider w.l.o.
Λ(0,x) =[
Λ1(x) 00 Λ2(x)
]
Λ1(x) and Λ2(x) are the speeds of propagation of z= [z1 z2]T . According to their signs:
homodirectional heterodirectional∀x∈ [0,1], Λ1(x)Λ2(x)>>> 0 ∀x∈ [0,1], Λ1(x)Λ2(x)<<< 0
Heterodirectional −→ one boundary condition on each side.
Homodirectional −→ two boundary conditions on the same side.
![Page 6: Local Exponential Stabilization of a 2 x 2 Quasilinear ...fa/cdps/talks/Vazquez.pdfLocal Exponential Stabilization of a 2 x 2 Quasilinear Hyperbolic System using Backstepping Rafael](https://reader030.vdocument.in/reader030/viewer/2022040703/5dd0bf73d6be591ccb62807f/html5/thumbnails/6.jpg)
Examples of Homo directional Systems
• road traffic: Aw-Rascle model
• heat exchanger
• plug-flow chemical reactor
• population dynamics (Lotka-Volterra) in laser chambers
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Examples of Hetero directional Systems
• Saint-Venant model of water waves in a channel
• gas flow in pipes
• cardiovascular flow in flexible blood vessels
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The control problem (hetero case)
zt +Λ(z,x)zx+ f (z,x) = 0
with boundary conditions
z1(0, t) = qz2(0, t) , q 6= 0
z2(1, t) = U(t) = actuation
Assume f (0,x) = 0 (equilibrium at the origin)
Task: find fbk law for U(t) to make z≡ 0 locally exponentially stable
(1) Stabilize linearized system using backstepping(2) Prove local stability for nonlinear system
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The control problem (hetero case)
zt +Λ(z,x)zx+ f (z,x) = 0
with boundary conditions
z1(0, t) = qz2(0, t) , q 6= 0
z2(1, t) = U(t) = actuation
Assume f (0,x) = 0 (equilibrium at the origin)
Task: find fbk law for U(t) to make z≡ 0 locally exponentially stable
(1) Stabilize linearized system using backstepping(2) Prove local stability for nonlinear system
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The control problem (hetero case)
zt +Λ(z,x)zx+ f (z,x) = 0
with boundary conditions
z1(0, t) = qz2(0, t) , q 6= 0
z2(1, t) = U(t) = actuation
Assume f (0,x) = 0 (equilibrium at the origin)
Task: find fbk law for U(t) to make z≡ 0 locally exponentially stable
Approach: (1) Stabilize linearized system using backstepping(2) Prove local stability for nonlinear system
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The linear case
ut = – ε1(x)ux+c1(x)v
vt = ε2(x)vx+c2(x)u
x∈ [0,1], ε1(x),ε2(x)> 0
with boundary conditions
u(t,0) = qv(t,0)
v(t,1) = U(t)
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Key Issue
v-PDE
u-PDEu(x,t)
v(x,t)
c₂c₁
A continuum of 1st-order (in time) subsystems with (potentially) positive feedback
coupling and small gain condition violated.
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Target system
αt = −ε1(x)αx
βt = ε2(x)βx
with boundary conditions
α(t,0) = qβ(t,0)β(t,1) = 0
Feedback connection severed throughout the domain, using control only at one boundary.
Cascade of two exp. stable transport PDEs (β → α).
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Backstepping transformation
α(t,x) = u(t,x)−∫ x
0Kuu(x,ξ)u(t,ξ)dξ−
∫ x
0Kuv(x,ξ)v(t,ξ)dξ
β(t,x) = v(t,x)−∫ x
0Kvu(x,ξ)u(t,ξ)dξ−
∫ x
0Kvv(x,ξ)v(t,ξ)dξ
Control law (set β(t,1) = 0)
U(t) =∫ 1
0Kvu(1,ξ)u(t,ξ)dξ+
∫ 1
0Kvv(1,ξ)v(t,ξ)dξ
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Backstepping transformation
α(t,x) = u(t,x)−∫ x
0Kuu(x,ξ)u(t,ξ)dξ−
∫ x
0Kuv(x,ξ)v(t,ξ)dξ
β(t,x) = v(t,x)−∫ x
0Kvu(x,ξ)u(t,ξ)dξ−
∫ x
0Kvv(x,ξ)v(t,ξ)dξ
Control law (set β(t,1) = 0)
U(t) =∫ 1
0Kvu(1,ξ)u(t,ξ)dξ+
∫ 1
0Kvv(1,ξ)v(t,ξ)dξ
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Kernel PDEs
First, for Kuu and Kuv:(
ε1(x)∂x+ ε1(ξ)∂ξ
)
Kuu = −ε′1(ξ)Kuu−c2(ξ)Kuv
(
ε1(x)∂x− ε2(ξ)∂ξ
)
Kuv = ε′2(ξ)Kuv−c1(ξ)Kuu,
with boundary conditions
Kuu(x,0) =ε2(0)qε1(0)
Kuv(x,0)
Kuv(x,x) =c1(x)
ε1(x)+ ε2(x).
A 2× 2 system of first-order linear hyperbolic PDE that evolves in the triangular domain
T = {(x,ξ) : 0≤ ξ ≤ x≤ 1}.
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Kernel PDEs
First, for Kuu and Kuv:(
ε1(x)∂x+ ε1(ξ)∂ξ
)
Kuu = −ε′1(ξ)Kuu−c2(ξ)Kuv
(
ε1(x)∂x− ε2(ξ)∂ξ
)
Kuv = ε′2(ξ)Kuv−c1(ξ)Kuu,
with boundary conditions
Kuu(x,0) =ε2(0)qε1(0)
Kuv(x,0)
Kuv(x,x) =c1(x)
ε1(x)+ ε2(x).
A 2× 2 system of first-order linear hyperbolic PDE that evolves in the triangular domain
T = {(x,ξ) : 0≤ ξ ≤ x≤ 1}.
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Second, for Kvu and Kvv:(
ε2(x)∂x+ ε2(ξ)∂ξ
)
Kvv = −ε′2(ξ)Kvv+c1(ξ)Kvu,
(
ε2(x)∂x− ε1(ξ)∂ξ
)
Kvu = ε′1(ξ)Kvu+c2(ξ)Kvv,
with boundary conditions
Kvv(x,0) =qε1(0)ε2(0)
Kvu(x,0)
Kvu(x,x) = −c2(x)
ε1(x)+ ε2(x)
Uncoupled with the previous PDE.
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Second, for Kvu and Kvv:(
ε2(x)∂x+ ε2(ξ)∂ξ
)
Kvv = −ε′2(ξ)Kvv+c1(ξ)Kvu,
(
ε2(x)∂x− ε1(ξ)∂ξ
)
Kvu = ε′1(ξ)Kvu+c2(ξ)Kvv,
with boundary conditions
Kvv(x,0) =qε1(0)ε2(0)
Kvu(x,0)
Kvu(x,x) = −c2(x)
ε1(x)+ ε2(x)
Uncoupled with the previous PDE.
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Linear example: constant coefficients
Benchmark system
ut +ux = ωv
vt −vx = ωu
with boundary conditions u(t,0) = qv(t,0), v(t,1) =U(t)
This 2×2 converts into one wave PDE with “anti-stiffness”
vtt = vxx+ω2v
Open-loop unstable for large ω.
For large enough ω no choices of k1, k2 in static output fbk law U = k1u(t,0)+ k2u(t,1)can achieve stability.
Recall the backstepping controller
U(t) =∫ 1
0Kvu(1,ξ)u(t,ξ)dξ+
∫ 1
0Kvv(1,ξ)v(t,ξ)dξ
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Linear example: constant coefficients
Benchmark system
ut +ux = ωv
vt −vx = ωu
with boundary conditions u(t,0) = qv(t,0), v(t,1) =U(t)
This 2×2 converts into one wave PDE with “anti-stiffness”
vtt = vxx+ω2v
Open-loop unstable for large ω.
For large enough ω no choices of k1, k2 in static output fbk law U = k1u(t,0)+ k2u(t,1)can achieve stability.
Recall the backstepping controller
U(t) =∫ 1
0Kvu(1,ξ)u(t,ξ)dξ+
∫ 1
0Kvv(1,ξ)v(t,ξ)dξ
![Page 22: Local Exponential Stabilization of a 2 x 2 Quasilinear ...fa/cdps/talks/Vazquez.pdfLocal Exponential Stabilization of a 2 x 2 Quasilinear Hyperbolic System using Backstepping Rafael](https://reader030.vdocument.in/reader030/viewer/2022040703/5dd0bf73d6be591ccb62807f/html5/thumbnails/22.jpg)
Linear example: constant coefficients
Benchmark system
ut +ux = ωv
vt −vx = ωu
with boundary conditions u(t,0) = qv(t,0), v(t,1) =U(t)
This 2×2 converts into one wave PDE with “anti-stiffness”
vtt = vxx+ω2v
Open-loop unstable for large ω.
For large enough ω no choices of k1, k2 in static output fbk law U = k1u(t,0)+ k2u(t,1)can achieve stability.
Recall the backstepping controller
U(t) =∫ 1
0Kvu(1,ξ)u(t,ξ)dξ+
∫ 1
0Kvv(1,ξ)v(t,ξ)dξ
![Page 23: Local Exponential Stabilization of a 2 x 2 Quasilinear ...fa/cdps/talks/Vazquez.pdfLocal Exponential Stabilization of a 2 x 2 Quasilinear Hyperbolic System using Backstepping Rafael](https://reader030.vdocument.in/reader030/viewer/2022040703/5dd0bf73d6be591ccb62807f/html5/thumbnails/23.jpg)
Control gain kernel Kvu(1,ξ) for q= 0.5 and ω = 1–10
w ! 1
w ! 2
w ! 3
w ! 4
w ! 5
w ! 6
w ! 7
w ! 8
w ! 9
w ! 10
0.2 0.4 0.6 0.8 1.0
Ξ
1
10
100
1000
104
#Kvu!Ξ"
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Control gain kernel Kvv(1,ξ) for q= 0.5 and ω = 1–10
w ! 1
w ! 2
w ! 3
w ! 4
w ! 1
w ! 5
w ! 6
w ! 7
w ! 8
w ! 9
w ! 10
0.2 0.4 0.6 0.8 1.0
Ξ
1
10
100
1000
104
#Kvv!Ξ"
![Page 25: Local Exponential Stabilization of a 2 x 2 Quasilinear ...fa/cdps/talks/Vazquez.pdfLocal Exponential Stabilization of a 2 x 2 Quasilinear Hyperbolic System using Backstepping Rafael](https://reader030.vdocument.in/reader030/viewer/2022040703/5dd0bf73d6be591ccb62807f/html5/thumbnails/25.jpg)
Note the log scale. The growth in ω seems exponential.
Control v(t,1) puts a strong emphasis on u(t,0.2) and v(t,0.3) — highly non-collocated!
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Inverse backstepping transformation
u(t,x) = α(t,x)+∫ x
0Lαα(x,ξ)α(t,ξ)dξ+
∫ x
0Lαβ(x,ξ)β(t,ξ)dξ,
v(t,x) = β(t,x)+∫ x
0Lβα(x,ξ)α(t,ξ)dξ+
∫ x
0Lββ(x,ξ)β(t,ξ)dξ,
One gets again four PDEs:(
ε1(x)∂x+ ε1(ξ)∂ξ
)
Lαα = −ε′1(ξ)Lαα+c1(x)L
βα,(
ε1(x)∂x− ε2(ξ)∂ξ
)
Lαβ = ε′2(ξ)Lαβ+c1(x)L
ββ,(
ε2(x)∂x− ε1(ξ)∂ξ
)
Lβα = ε′1(ξ)Lβα−c2(x)L
αα(
ε2(x)∂x+ ε2(ξ)∂ξ
)
Lββ = −ε′2(ξ)Lββ−c2(x)L
αβ
with boundary conditions
Lαα(x,0) =ε2(0)qε1(0)
Lαβ(x,0), Lαβ(x,x) =c1(x)
ε1(x)+ ε2(x)
Lβα(x,x) = −c2(x)
ε1(x)+ ε2(x), Lββ(x,0) =
qε1(0)ε2(0)
Lβα(x,0)
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Kernel well-posedness
Consider the following “generalized Goursat problem” of which the direct and inverse ker-
nel equations are a particular case:
(
ε1(x)∂x+ ε1(ξ)∂ξ
)
F1 = g1(x,ξ)+4
∑i=1
C1i(x,ξ)F i(x,ξ),
(
ε1(x)∂x− ε2(ξ)∂ξ
)
F2 = g2(x,ξ)+4
∑i=1
C2i(x,ξ)F i(x,ξ),
(
ε2(x)∂x− ε1(ξ)∂ξ
)
F3 = g3(x,ξ)+4
∑i=1
C3i(x,ξ)F i(x,ξ),
(
ε2(x)∂x+ ε2(ξ)∂ξ
)
F4 = g4(x,ξ)+4
∑i=1
C4i(x,ξ)F i(x,ξ),
with boundary conditions
F1(x,0) = h1(x)+q1(x)F2(x,0)+q2(x)F
3(x,0),
F2(x,x) = h2(x), F3(x,x)= h3(x),
F4(x,0) = h4(x)+q3(x)F2(x,0)+q4(x)F
3(x,0).
evolving in the domain T = {(x,ξ) : 0≤ ξ ≤ x≤ 1}.
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Transforming the PDEs into integral equations along the characteristic lines, and using the
method of successive approximations, we prove:
Theorem Under the assumptions
qi,hi ∈ C ([0,1]), gi,Cji ∈ C (T ), i, j = 1,2,3,4
and ε1,ε2 ∈ C ([0,1]) with ε1(x),ε2(x) > 0, there exists a unique C (T ) solution F i, i =
1,2,3,4.
Theorem Under the additional assumptions
εi,qi,hi ∈ CN([0,1]), gi,Cji ∈ C
N(T ),
there exists a unique C N(T ) solution F i , i = 1,2,3,4.
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Transforming the PDEs into integral equations along the characteristic lines, and using the
method of successive approximations, we prove:
Theorem Under the assumptions
qi,hi ∈ C ([0,1]), gi,Cji ∈ C (T ), i, j = 1,2,3,4
and ε1,ε2 ∈ C ([0,1]) with ε1(x),ε2(x) > 0, there exists a unique C (T ) solution F i, i =
1,2,3,4.
Theorem Under the additional assumptions
εi,qi,hi ∈ CN([0,1]), gi,Cji ∈ C
N(T ),
there exists a unique C N(T ) solution F i , i = 1,2,3,4.
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Transforming the PDEs into integral equations along the characteristic lines, and using the
method of successive approximations, we prove:
Theorem Under the assumptions
qi,hi ∈ C ([0,1]), gi,Cji ∈ C (T ), i, j = 1,2,3,4
and ε1,ε2 ∈ C ([0,1]) with ε1(x),ε2(x) > 0, there exists a unique C (T ) solution F i, i =
1,2,3,4.
Theorem Under the additional assumptions
εi,qi,hi ∈ CN([0,1]), gi,Cji ∈ C
N(T ),
there exists a unique C N(T ) solution F i , i = 1,2,3,4.
![Page 31: Local Exponential Stabilization of a 2 x 2 Quasilinear ...fa/cdps/talks/Vazquez.pdfLocal Exponential Stabilization of a 2 x 2 Quasilinear Hyperbolic System using Backstepping Rafael](https://reader030.vdocument.in/reader030/viewer/2022040703/5dd0bf73d6be591ccb62807f/html5/thumbnails/31.jpg)
An observer-based controller with sensing of u(1, t)
ut = −ε1ux+c1(x)v− ε1Puu(x,1)(u(t,1)− u(t,1))
vt = ε2vx+c2(x)u− ε1Pvu(x,1)(u(t,1)− u(t,1))
u(t,0) = qv(t,0)
v(t,1) = U(t) =∫ 1
0Kvu(1,ξ)u(t,ξ)dξ+
∫ 1
0Kvv(1,ξ)v(t,ξ)dξ
Observer gains obtained from(ε1(x)∂x+ ε1(ξ)∂ξ
)Puu = −ε′1(ξ)P
uu−c1(x)Pvu
(ε2(x)∂x− ε1(ξ)∂ξ
)Pvu = ε′1(ξ)P
vu+c2(x)Puu
Puu(0,ξ) = qPvu(0,ξ), Pvu(x,x) =−c2(x)
ε1(x)+ ε2(x)
(ε1(x)∂x− ε2(ξ)∂ξ
)Puv = ε′2(ξ)P
uv−c1(x)Pvv
(ε2(x)∂x+ ε2(ξ)∂ξ
)Pvv = −ε′2(ξ)P
vv+c2(x)Puv
Pvv(0,ξ) =1q
Puv(0,ξ), Puv(x,x) =c1(x)
ε1(x)+ ε2(x)
![Page 32: Local Exponential Stabilization of a 2 x 2 Quasilinear ...fa/cdps/talks/Vazquez.pdfLocal Exponential Stabilization of a 2 x 2 Quasilinear Hyperbolic System using Backstepping Rafael](https://reader030.vdocument.in/reader030/viewer/2022040703/5dd0bf73d6be591ccb62807f/html5/thumbnails/32.jpg)
An observer-based controller with sensing of u(1, t)
ut = −ε1ux+c1(x)v− ε1Puu(x,1)(u(t,1)− u(t,1))
vt = ε2vx+c2(x)u− ε1Pvu(x,1)(u(t,1)− u(t,1))
u(t,0) = qv(t,0)
v(t,1) = U(t) =∫ 1
0Kvu(1,ξ)u(t,ξ)dξ+
∫ 1
0Kvv(1,ξ)v(t,ξ)dξ
Observer gains obtained from(ε1(x)∂x+ ε1(ξ)∂ξ
)Puu = −ε′1(ξ)P
uu−c1(x)Pvu
(ε2(x)∂x− ε1(ξ)∂ξ
)Pvu = ε′1(ξ)P
vu+c2(x)Puu
Puu(0,ξ) = qPvu(0,ξ), Pvu(x,x) =−c2(x)
ε1(x)+ ε2(x)
(ε1(x)∂x− ε2(ξ)∂ξ
)Puv = ε′2(ξ)P
uv−c1(x)Pvv
(ε2(x)∂x+ ε2(ξ)∂ξ
)Pvv = −ε′2(ξ)P
vv+c2(x)Puv
Pvv(0,ξ) =1q
Puv(0,ξ), Puv(x,x) =c1(x)
ε1(x)+ ε2(x)
![Page 33: Local Exponential Stabilization of a 2 x 2 Quasilinear ...fa/cdps/talks/Vazquez.pdfLocal Exponential Stabilization of a 2 x 2 Quasilinear Hyperbolic System using Backstepping Rafael](https://reader030.vdocument.in/reader030/viewer/2022040703/5dd0bf73d6be591ccb62807f/html5/thumbnails/33.jpg)
An observer-based controller with sensing of u(1, t)
ut = −ε1ux+c1(x)v− ε1Puu(x,1)(u(t,1)− u(t,1))
vt = ε2vx+c2(x)u− ε1Pvu(x,1)(u(t,1)− u(t,1))
u(t,0) = qv(t,0)
v(t,1) = U(t) =∫ 1
0Kvu(1,ξ)u(t,ξ)dξ+
∫ 1
0Kvv(1,ξ)v(t,ξ)dξ
Observer gains obtained from(ε1(x)∂x+ ε1(ξ)∂ξ
)Puu = −ε′1(ξ)P
uu−c1(x)Pvu
(ε2(x)∂x− ε1(ξ)∂ξ
)Pvu = ε′1(ξ)P
vu+c2(x)Puu
Puu(0,ξ) = qPvu(0,ξ), Pvu(x,x) =−c2(x)
ε1(x)+ ε2(x)
(ε1(x)∂x− ε2(ξ)∂ξ
)Puv = ε′2(ξ)P
uv−c1(x)Pvv
(ε2(x)∂x+ ε2(ξ)∂ξ
)Pvv = −ε′2(ξ)P
vv+c2(x)Puv
Pvv(0,ξ) =1q
Puv(0,ξ), Puv(x,x) =c1(x)
ε1(x)+ ε2(x)
![Page 34: Local Exponential Stabilization of a 2 x 2 Quasilinear ...fa/cdps/talks/Vazquez.pdfLocal Exponential Stabilization of a 2 x 2 Quasilinear Hyperbolic System using Backstepping Rafael](https://reader030.vdocument.in/reader030/viewer/2022040703/5dd0bf73d6be591ccb62807f/html5/thumbnails/34.jpg)
Back to the Nonlinear Case
zt +Λ(z,x)zx+ f (z,x) = 0
z1(0, t) = qz2(0, t), z2(1, t) =U(t)
Consider only the state-fbk problem here, but output-fbk also possible.
Compute
∂ f∂z
(z,x)
∣∣∣∣z=0
=
[f11(x) f12(x)f21(x) f22(x)
]
Define
ϕ1(x) = exp
(∫ x
0
f11(s)Λ1(s)
ds
)
ϕ2(x) = exp
(
−
∫ x
0
f22(s)Λ2(s)
ds
)
![Page 35: Local Exponential Stabilization of a 2 x 2 Quasilinear ...fa/cdps/talks/Vazquez.pdfLocal Exponential Stabilization of a 2 x 2 Quasilinear Hyperbolic System using Backstepping Rafael](https://reader030.vdocument.in/reader030/viewer/2022040703/5dd0bf73d6be591ccb62807f/html5/thumbnails/35.jpg)
Back to the Nonlinear Case
zt +Λ(z,x)zx+ f (z,x) = 0
z1(0, t) = qz2(0, t), z2(1, t) =U(t)
Consider only the state-fbk problem here, but output-fbk also possible.
Compute
∂ f∂z
(z,x)
∣∣∣∣z=0
=
[f11(x) f12(x)f21(x) f22(x)
]
Define
ϕ1(x) = exp
(∫ x
0
f11(s)Λ1(s)
ds
)
ϕ2(x) = exp
(
−
∫ x
0
f22(s)Λ2(s)
ds
)
![Page 36: Local Exponential Stabilization of a 2 x 2 Quasilinear ...fa/cdps/talks/Vazquez.pdfLocal Exponential Stabilization of a 2 x 2 Quasilinear Hyperbolic System using Backstepping Rafael](https://reader030.vdocument.in/reader030/viewer/2022040703/5dd0bf73d6be591ccb62807f/html5/thumbnails/36.jpg)
In re-scaled variables
w=
[ϕ1(x) 0
0 ϕ2(x)
]
z= Φ(x)z
the system is
wt −Σ(x)wx−C(x)w︸ ︷︷ ︸
design bks contr for this syst
+ΛNL(w,x)wx+ fNL(w,x)︸ ︷︷ ︸
nonlinear perturbations
= 0
where
Σ(x) =−Λ(0,x) =[−Λ1(x) 0
0 −Λ2(x)
]
, C(x) =
[0 − f12
− f21 0
]
Control law in z variables:
U = ϕ2(1)∫ 1
0
Kvu(1,ξ)ϕ1(ξ)
z1(ξ, t)dξ+ϕ2(1)∫ 1
0
Kvv(1,ξ)ϕ2(ξ)
z2(ξ, t)dξ
![Page 37: Local Exponential Stabilization of a 2 x 2 Quasilinear ...fa/cdps/talks/Vazquez.pdfLocal Exponential Stabilization of a 2 x 2 Quasilinear Hyperbolic System using Backstepping Rafael](https://reader030.vdocument.in/reader030/viewer/2022040703/5dd0bf73d6be591ccb62807f/html5/thumbnails/37.jpg)
In re-scaled variables
w=
[ϕ1(x) 0
0 ϕ2(x)
]
z= Φ(x)z
the system is
wt −Σ(x)wx−C(x)w︸ ︷︷ ︸
design bks contr for this syst
+ΛNL(w,x)wx+ fNL(w,x)︸ ︷︷ ︸
nonlinear perturbations
= 0
where
Σ(x) =−Λ(0,x) =[−Λ1(x) 0
0 −Λ2(x)
]
, C(x) =
[0 − f12
− f21 0
]
Control law in z variables:
U = ϕ2(1)∫ 1
0
Kvu(1,ξ)ϕ1(ξ)
z1(ξ, t)dξ+ϕ2(1)∫ 1
0
Kvv(1,ξ)ϕ2(ξ)
z2(ξ, t)dξ
![Page 38: Local Exponential Stabilization of a 2 x 2 Quasilinear ...fa/cdps/talks/Vazquez.pdfLocal Exponential Stabilization of a 2 x 2 Quasilinear Hyperbolic System using Backstepping Rafael](https://reader030.vdocument.in/reader030/viewer/2022040703/5dd0bf73d6be591ccb62807f/html5/thumbnails/38.jpg)
In re-scaled variables
w=
[ϕ1(x) 0
0 ϕ2(x)
]
z= Φ(x)z
the system is
wt −Σ(x)wx−C(x)w︸ ︷︷ ︸
design bks contr for this syst
+ΛNL(w,x)wx+ fNL(w,x)︸ ︷︷ ︸
nonlinear perturbations
= 0
where
Σ(x) =−Λ(0,x) =[−Λ1(x) 0
0 −Λ2(x)
]
, C(x) =
[0 − f12
− f21 0
]
Control law in z variables:
U = ϕ2(1)∫ 1
0
Kvu(1,ξ)ϕ1(ξ)
z1(ξ, t)dξ+ϕ2(1)∫ 1
0
Kvv(1,ξ)ϕ2(ξ)
z2(ξ, t)dξ
![Page 39: Local Exponential Stabilization of a 2 x 2 Quasilinear ...fa/cdps/talks/Vazquez.pdfLocal Exponential Stabilization of a 2 x 2 Quasilinear Hyperbolic System using Backstepping Rafael](https://reader030.vdocument.in/reader030/viewer/2022040703/5dd0bf73d6be591ccb62807f/html5/thumbnails/39.jpg)
Proof of stability
For linear problem, exp stability in L2.
For nonlinear case, local exp stability in H2.
Proof is involved. Needs a careful selection of Lyapunov functions (complications due to
the ΛNL(w,x)wx term).
Direct and inverse bkst transforms applied to obtain the system + perturbation term in
target variables.
To go to H1 and H2, take t-derivatives instead of x-derivatives to simplify the process.
![Page 40: Local Exponential Stabilization of a 2 x 2 Quasilinear ...fa/cdps/talks/Vazquez.pdfLocal Exponential Stabilization of a 2 x 2 Quasilinear Hyperbolic System using Backstepping Rafael](https://reader030.vdocument.in/reader030/viewer/2022040703/5dd0bf73d6be591ccb62807f/html5/thumbnails/40.jpg)
Proof of stability
For linear problem, exp stability in L2.
For nonlinear case, local exp stability in H2.
Proof is involved. Needs a careful selection of Lyapunov functions (complications due to
the ΛNL(w,x)wx term).
Direct and inverse bkst transforms applied to obtain the system + perturbation term in
target variables.
To go to H1 and H2, take t-derivatives instead of x-derivatives to simplify the process.
![Page 41: Local Exponential Stabilization of a 2 x 2 Quasilinear ...fa/cdps/talks/Vazquez.pdfLocal Exponential Stabilization of a 2 x 2 Quasilinear Hyperbolic System using Backstepping Rafael](https://reader030.vdocument.in/reader030/viewer/2022040703/5dd0bf73d6be591ccb62807f/html5/thumbnails/41.jpg)
Proof of stability
For linear problem, exp stability in L2.
For nonlinear case, local exp stability in H2.
Proof is involved. Needs a careful selection of Lyapunov functions (complications due to
the ΛNL(w,x)wx term).
Direct and inverse bkst transforms applied to obtain the system + perturbation term in
target variables.
To go to H1 and H2, take t-derivatives instead of x-derivatives to simplify the process.
![Page 42: Local Exponential Stabilization of a 2 x 2 Quasilinear ...fa/cdps/talks/Vazquez.pdfLocal Exponential Stabilization of a 2 x 2 Quasilinear Hyperbolic System using Backstepping Rafael](https://reader030.vdocument.in/reader030/viewer/2022040703/5dd0bf73d6be591ccb62807f/html5/thumbnails/42.jpg)
Proof of stability
For linear problem, exp stability in L2.
For nonlinear case, local exp stability in H2.
Proof is involved. Needs a careful selection of Lyapunov functions (complications due to
the ΛNL(w,x)wx term).
Direct and inverse bkst transforms applied to obtain the system + perturbation term in
target variables.
To go to H1 and H2, take t-derivatives instead of x-derivatives to simplify the process.
![Page 43: Local Exponential Stabilization of a 2 x 2 Quasilinear ...fa/cdps/talks/Vazquez.pdfLocal Exponential Stabilization of a 2 x 2 Quasilinear Hyperbolic System using Backstepping Rafael](https://reader030.vdocument.in/reader030/viewer/2022040703/5dd0bf73d6be591ccb62807f/html5/thumbnails/43.jpg)
Proof of stability
For linear problem, exp stability in L2.
For nonlinear case, local exp stability in H2.
Proof is involved. Needs a careful selection of Lyapunov functions (complications due to
the ΛNL(w,x)wx term).
Direct and inverse bkst transforms applied to obtain the system + perturbation term in
target variables.
To go to H1 and H2, take t-derivatives instead of x-derivatives to simplify the process.
![Page 44: Local Exponential Stabilization of a 2 x 2 Quasilinear ...fa/cdps/talks/Vazquez.pdfLocal Exponential Stabilization of a 2 x 2 Quasilinear Hyperbolic System using Backstepping Rafael](https://reader030.vdocument.in/reader030/viewer/2022040703/5dd0bf73d6be591ccb62807f/html5/thumbnails/44.jpg)
L2 analysis step
System written in target variables γ = [α β]T
γt −Σ(x)γx︸ ︷︷ ︸
linear stable part
+ F3[γ,γx]+F4[γ]︸ ︷︷ ︸
nonlinear perturbation
= 0,
F3 and F4 are functionals in terms of backstepping kernels. Boundary conditions:
α(0, t) = qβ(0, t)β(1, t) = 0
Lyapunov function:
U =
∫ 1
0γT(x, t)D(x)γ(x, t)dx, D(x) =
λ1
eµ(1−x)
Λ1(x)0
0(
q2λ1eµ+λ2
)eµx
Λ2(x)
with µ= λ1maxx∈[0,1]{
1Λ1(x)
, 1Λ2(x)
}
and choosing λ1,λ2 > 0. Then if ‖γ‖∞ < δ1
U ≤ −λ1U −λ2
(
α2(1, t)+β2(0, t))
+C1U3/2+C2‖γx‖∞U,
![Page 45: Local Exponential Stabilization of a 2 x 2 Quasilinear ...fa/cdps/talks/Vazquez.pdfLocal Exponential Stabilization of a 2 x 2 Quasilinear Hyperbolic System using Backstepping Rafael](https://reader030.vdocument.in/reader030/viewer/2022040703/5dd0bf73d6be591ccb62807f/html5/thumbnails/45.jpg)
L2 analysis step
System written in target variables γ = [α β]T
γt −Σ(x)γx︸ ︷︷ ︸
linear stable part
+ F3[γ,γx]+F4[γ]︸ ︷︷ ︸
nonlinear perturbation
= 0,
F3 and F4 are functionals in terms of backstepping kernels. Boundary conditions:
α(0, t) = qβ(0, t)β(1, t) = 0
Lyapunov function:
U =
∫ 1
0γT(x, t)D(x)γ(x, t)dx, D(x) =
λ1
eµ(1−x)
Λ1(x)0
0(
q2λ1eµ+λ2
)eµx
Λ2(x)
with µ= λ1maxx∈[0,1]{
1Λ1(x)
, 1Λ2(x)
}
and choosing λ1,λ2 > 0.Then if ‖γ‖∞ < δ1
U ≤ −λ1U −λ2
(
α2(1, t)+β2(0, t))
+C1U3/2+C2‖γx‖∞U,
![Page 46: Local Exponential Stabilization of a 2 x 2 Quasilinear ...fa/cdps/talks/Vazquez.pdfLocal Exponential Stabilization of a 2 x 2 Quasilinear Hyperbolic System using Backstepping Rafael](https://reader030.vdocument.in/reader030/viewer/2022040703/5dd0bf73d6be591ccb62807f/html5/thumbnails/46.jpg)
L2 analysis step
System written in target variables γ = [α β]T
γt −Σ(x)γx︸ ︷︷ ︸
linear stable part
+ F3[γ,γx]+F4[γ]︸ ︷︷ ︸
nonlinear perturbation
= 0,
F3 and F4 are functionals in terms of backstepping kernels. Boundary conditions:
α(0, t) = qβ(0, t)β(1, t) = 0
Lyapunov function:
U =
∫ 1
0γT(x, t)D(x)γ(x, t)dx, D(x) =
λ1
eµ(1−x)
Λ1(x)0
0(
q2λ1eµ+λ2
)eµx
Λ2(x)
with µ= λ1maxx∈[0,1]{
1Λ1(x)
, 1Λ2(x)
}
and choosing λ1,λ2 > 0. Then if ‖γ‖∞ < δ1
U ≤ −λ1U −λ2
(
α2(1, t)+β2(0, t))
+C1U3/2+C2‖γx‖∞U,
![Page 47: Local Exponential Stabilization of a 2 x 2 Quasilinear ...fa/cdps/talks/Vazquez.pdfLocal Exponential Stabilization of a 2 x 2 Quasilinear Hyperbolic System using Backstepping Rafael](https://reader030.vdocument.in/reader030/viewer/2022040703/5dd0bf73d6be591ccb62807f/html5/thumbnails/47.jpg)
H1 analysis step
γtt −Σ(x)γtx︸ ︷︷ ︸
linear stable part
+F1[γ]γtx+F5[γ,γx,γt]+F6[γ,γt]︸ ︷︷ ︸
nonlinear perturbation
= 0,
Lyapunov function:
V =∫ 1
0γTt (x, t)R[γ](x)γt(x, t)dx, R[γ](x) = D(x)+
[0 ψ[γ]
ψ[γ] 0
]
where ψ[γ] = D11(x)(F1[γ])12−D22(x)(F1[γ])21ε2(x)+ε1(x)+(F1[γ])11−(F1[γ])22
. Then if ‖γ‖∞ < δ2
V ≤ −λ3V −λ4
(
α2t (1, t)+β2
t (0, t))
+C3V‖‖∞
![Page 48: Local Exponential Stabilization of a 2 x 2 Quasilinear ...fa/cdps/talks/Vazquez.pdfLocal Exponential Stabilization of a 2 x 2 Quasilinear Hyperbolic System using Backstepping Rafael](https://reader030.vdocument.in/reader030/viewer/2022040703/5dd0bf73d6be591ccb62807f/html5/thumbnails/48.jpg)
H1 analysis step
γtt −Σ(x)γtx︸ ︷︷ ︸
linear stable part
+F1[γ]γtx+F5[γ,γx,γt]+F6[γ,γt]︸ ︷︷ ︸
nonlinear perturbation
= 0,
Lyapunov function:
V =
∫ 1
0γTt (x, t)R[γ](x)γt(x, t)dx, R[γ](x) = D(x)+
[0 ψ[γ]
ψ[γ] 0
]
where ψ[γ] = D11(x)(F1[γ])12−D22(x)(F1[γ])21ε2(x)+ε1(x)+(F1[γ])11−(F1[γ])22
.Then if ‖γ‖∞ < δ2
V ≤ −λ3V −λ4
(
α2t (1, t)+β2
t (0, t))
+C3V‖γt‖∞
![Page 49: Local Exponential Stabilization of a 2 x 2 Quasilinear ...fa/cdps/talks/Vazquez.pdfLocal Exponential Stabilization of a 2 x 2 Quasilinear Hyperbolic System using Backstepping Rafael](https://reader030.vdocument.in/reader030/viewer/2022040703/5dd0bf73d6be591ccb62807f/html5/thumbnails/49.jpg)
H1 analysis step
γtt −Σ(x)γtx︸ ︷︷ ︸
linear stable part
+F1[γ]γtx+F5[γ,γx,γt]+F6[γ,γt]︸ ︷︷ ︸
nonlinear perturbation
= 0,
Lyapunov function:
V =
∫ 1
0γTt (x, t)R[γ](x)γt(x, t)dx, R[γ](x) = D(x)+
[0 ψ[γ]
ψ[γ] 0
]
where ψ[γ] = D11(x)(F1[γ])12−D22(x)(F1[γ])21ε2(x)+ε1(x)+(F1[γ])11−(F1[γ])22
. Then if ‖γ‖∞ < δ2
V ≤ −λ3V −λ4
(
α2t (1, t)+β2
t (0, t))
+C3V‖γt‖∞
![Page 50: Local Exponential Stabilization of a 2 x 2 Quasilinear ...fa/cdps/talks/Vazquez.pdfLocal Exponential Stabilization of a 2 x 2 Quasilinear Hyperbolic System using Backstepping Rafael](https://reader030.vdocument.in/reader030/viewer/2022040703/5dd0bf73d6be591ccb62807f/html5/thumbnails/50.jpg)
H2 analysis
γttt −Σ(x)γttx︸ ︷︷ ︸
linear stable part
+F1[γ]γttx+F7[γ,γx,γt ,γtx,γtt]+F8[γ,γt,γtt ]︸ ︷︷ ︸
nonlinear perturbation
= 0,
Lyapunov function:
W =∫ 1
0γTtt(x, t)R[γ](x)γtt(x, t)dx
Then if ‖γ‖∞+‖γt‖∞ < δ3
W ≤ −λ5W−λ6
(
α2tt(1, t)+β2
tt(0, t))
+C4WV1/2+C5VW1/2+C6W3/2
![Page 51: Local Exponential Stabilization of a 2 x 2 Quasilinear ...fa/cdps/talks/Vazquez.pdfLocal Exponential Stabilization of a 2 x 2 Quasilinear Hyperbolic System using Backstepping Rafael](https://reader030.vdocument.in/reader030/viewer/2022040703/5dd0bf73d6be591ccb62807f/html5/thumbnails/51.jpg)
H2 analysis
γttt −Σ(x)γttx︸ ︷︷ ︸
linear stable part
+F1[γ]γttx+F7[γ,γx,γt ,γtx,γtt]+F8[γ,γt,γtt ]︸ ︷︷ ︸
nonlinear perturbation
= 0,
Lyapunov function:
W =∫ 1
0γTtt(x, t)R[γ](x)γtt(x, t)dx
Then if ‖γ‖∞+‖γt‖∞ < δ3
W ≤ −λ5W−λ6
(
α2tt(1, t)+β2
tt(0, t))
+C4WV1/2+C5VW1/2+C6W3/2
![Page 52: Local Exponential Stabilization of a 2 x 2 Quasilinear ...fa/cdps/talks/Vazquez.pdfLocal Exponential Stabilization of a 2 x 2 Quasilinear Hyperbolic System using Backstepping Rafael](https://reader030.vdocument.in/reader030/viewer/2022040703/5dd0bf73d6be591ccb62807f/html5/thumbnails/52.jpg)
H2 analysis
γttt −Σ(x)γttx︸ ︷︷ ︸
linear stable part
+F1[γ]γttx+F7[γ,γx,γt ,γtx,γtt]+F8[γ,γt,γtt ]︸ ︷︷ ︸
nonlinear perturbation
= 0,
Lyapunov function:
W =∫ 1
0γTtt(x, t)R[γ](x)γtt(x, t)dx
Then if ‖γ‖∞+‖γt‖∞ < δ3
W ≤ −λ5W−λ6
(
α2tt(1, t)+β2
tt(0, t))
+C4WV1/2+C5VW1/2+C6W3/2
![Page 53: Local Exponential Stabilization of a 2 x 2 Quasilinear ...fa/cdps/talks/Vazquez.pdfLocal Exponential Stabilization of a 2 x 2 Quasilinear Hyperbolic System using Backstepping Rafael](https://reader030.vdocument.in/reader030/viewer/2022040703/5dd0bf73d6be591ccb62807f/html5/thumbnails/53.jpg)
To relate γt , γtt and the H1, H2 norms of γ use the following lemmas:
Lemma 1 . If ‖γ‖∞ < δ4
‖γt‖∞ ≤ c1(‖γx‖∞+‖γ‖∞)
‖γx‖∞ ≤ c3(‖γt‖∞+‖γ‖∞)
‖γt‖L2 ≤ c2(‖γx‖L2+‖γ‖L2
)
‖γx‖L2 ≤ c4(‖γt‖L2+‖γ‖L2
)
Lemma 2 . If ‖γ‖∞+‖γt‖∞ < δ5
‖γtt‖∞ ≤ c1(‖γxx‖∞+‖γx‖∞+‖γ‖∞)
‖γxx‖∞ ≤ c3(‖γtt‖∞+‖γt‖∞+‖γ‖∞)
‖γtt‖L2 ≤ c2(‖γxx‖L2+‖γx‖L2+‖γ‖L2
)
‖γxx‖L2 ≤ c4(‖γtt‖L2+‖γt‖L2+‖γ‖L2
)
![Page 54: Local Exponential Stabilization of a 2 x 2 Quasilinear ...fa/cdps/talks/Vazquez.pdfLocal Exponential Stabilization of a 2 x 2 Quasilinear Hyperbolic System using Backstepping Rafael](https://reader030.vdocument.in/reader030/viewer/2022040703/5dd0bf73d6be591ccb62807f/html5/thumbnails/54.jpg)
To relate γt , γtt and the H1, H2 norms of γ use the following lemmas:
Lemma 1 . If ‖γ‖∞ < δ4
‖γt‖∞ ≤ c1(‖γx‖∞+‖γ‖∞)
‖γx‖∞ ≤ c3(‖γt‖∞+‖γ‖∞)
‖γt‖L2 ≤ c2(‖γx‖L2+‖γ‖L2
)
‖γx‖L2 ≤ c4(‖γt‖L2+‖γ‖L2
)
Lemma 2 . If ‖γ‖∞+‖γt‖∞ < δ5
‖γtt‖∞ ≤ c1(‖γxx‖∞+‖γx‖∞+‖γ‖∞)
‖γxx‖∞ ≤ c3(‖γtt‖∞+‖γt‖∞+‖γ‖∞)
‖γtt‖L2 ≤ c2(‖γxx‖L2+‖γx‖L2+‖γ‖L2
)
‖γxx‖L2 ≤ c4(‖γtt‖L2+‖γt‖L2+‖γ‖L2
)
![Page 55: Local Exponential Stabilization of a 2 x 2 Quasilinear ...fa/cdps/talks/Vazquez.pdfLocal Exponential Stabilization of a 2 x 2 Quasilinear Hyperbolic System using Backstepping Rafael](https://reader030.vdocument.in/reader030/viewer/2022040703/5dd0bf73d6be591ccb62807f/html5/thumbnails/55.jpg)
To relate γt , γtt and the H1, H2 norms of γ use the following lemmas:
Lemma 1 . If ‖γ‖∞ < δ4
‖γt‖∞ ≤ c1(‖γx‖∞+‖γ‖∞)
‖γx‖∞ ≤ c3(‖γt‖∞+‖γ‖∞)
‖γt‖L2 ≤ c2(‖γx‖L2+‖γ‖L2
)
‖γx‖L2 ≤ c4(‖γt‖L2+‖γ‖L2
)
Lemma 2 . If ‖γ‖∞+‖γt‖∞ < δ5
‖γtt‖∞ ≤ c1(‖γxx‖∞+‖γx‖∞+‖γ‖∞)
‖γxx‖∞ ≤ c3(‖γtt‖∞+‖γt‖∞+‖γ‖∞)
‖γtt‖L2 ≤ c2(‖γxx‖L2+‖γx‖L2+‖γ‖L2
)
‖γxx‖L2 ≤ c4(‖γtt‖L2+‖γt‖L2+‖γ‖L2
)
![Page 56: Local Exponential Stabilization of a 2 x 2 Quasilinear ...fa/cdps/talks/Vazquez.pdfLocal Exponential Stabilization of a 2 x 2 Quasilinear Hyperbolic System using Backstepping Rafael](https://reader030.vdocument.in/reader030/viewer/2022040703/5dd0bf73d6be591ccb62807f/html5/thumbnails/56.jpg)
The nonlinear result
Define Lyap. fcn.
S=U +V +W
Combining previous results if ‖γ‖∞+‖γt‖∞ < δ
S≤−λS+CS3/2
for λ,C> 0.
Noting ‖γ‖∞+‖γt‖∞ ≤C7Sand that S is equivalent to the H2 norm of γ we obtain
Theorem [Vazquez, Coron, Krstic, 2011 CDC]
With the linear backstepping controller, ∃δ0,M0,γ0 > 0 such that
‖w0‖H2 ≤ δ0
⇓
‖w(·, t)‖H2 ≤ M0e−γ0t‖w0‖H2 .
![Page 57: Local Exponential Stabilization of a 2 x 2 Quasilinear ...fa/cdps/talks/Vazquez.pdfLocal Exponential Stabilization of a 2 x 2 Quasilinear Hyperbolic System using Backstepping Rafael](https://reader030.vdocument.in/reader030/viewer/2022040703/5dd0bf73d6be591ccb62807f/html5/thumbnails/57.jpg)
The nonlinear result
Define Lyap. fcn.
S=U +V +W
Combining previous results if ‖γ‖∞+‖γt‖∞ < δ
S≤−λS+CS3/2
for λ,C> 0.
Noting ‖γ‖∞+‖γt‖∞ ≤C7Sand that S is equivalent to the H2 norm of γ we obtain
Theorem [Vazquez, Coron, Krstic, 2011 CDC]
With the linear backstepping controller, ∃δ0,M0,γ0 > 0 such that
‖w0‖H2 ≤ δ0
⇓
‖w(·, t)‖H2 ≤ M0e−γ0t‖w0‖H2 .
![Page 58: Local Exponential Stabilization of a 2 x 2 Quasilinear ...fa/cdps/talks/Vazquez.pdfLocal Exponential Stabilization of a 2 x 2 Quasilinear Hyperbolic System using Backstepping Rafael](https://reader030.vdocument.in/reader030/viewer/2022040703/5dd0bf73d6be591ccb62807f/html5/thumbnails/58.jpg)
The nonlinear result
Define Lyap. fcn.
S=U +V +W
Combining previous results if ‖γ‖∞+‖γt‖∞ < δ
S≤−λS+CS3/2
for λ,C> 0.
Noting ‖γ‖∞+‖γt‖∞ ≤C7Sand that S is equivalent to the H2 norm of γ we obtain
Theorem [Vazquez, Coron, Krstic, 2011 CDC]
With the linear backstepping controller, ∃δ0,M0,γ0 > 0 such that
‖w0‖H2 ≤ δ0
⇓
‖w(·, t)‖H2 ≤ M0e−γ0t‖w0‖H2 .
![Page 59: Local Exponential Stabilization of a 2 x 2 Quasilinear ...fa/cdps/talks/Vazquez.pdfLocal Exponential Stabilization of a 2 x 2 Quasilinear Hyperbolic System using Backstepping Rafael](https://reader030.vdocument.in/reader030/viewer/2022040703/5dd0bf73d6be591ccb62807f/html5/thumbnails/59.jpg)
Summary
Backstepping removes restrictions that have plagued static output feedback designs for
hyperbolic 2×2 systems
In the nonlinear case, local stabilization in H2
Interesting open problem: N×N systems (slugging flows in offshore oil rigs)
![Page 60: Local Exponential Stabilization of a 2 x 2 Quasilinear ...fa/cdps/talks/Vazquez.pdfLocal Exponential Stabilization of a 2 x 2 Quasilinear Hyperbolic System using Backstepping Rafael](https://reader030.vdocument.in/reader030/viewer/2022040703/5dd0bf73d6be591ccb62807f/html5/thumbnails/60.jpg)
Summary
Backstepping removes restrictions that have plagued static output feedback designs for
hyperbolic 2×2 systems
In the nonlinear case, local stabilization in H2
Interesting open problem: N×N systems (slugging flows in offshore oil rigs)
![Page 61: Local Exponential Stabilization of a 2 x 2 Quasilinear ...fa/cdps/talks/Vazquez.pdfLocal Exponential Stabilization of a 2 x 2 Quasilinear Hyperbolic System using Backstepping Rafael](https://reader030.vdocument.in/reader030/viewer/2022040703/5dd0bf73d6be591ccb62807f/html5/thumbnails/61.jpg)
Summary
Backstepping removes restrictions that have plagued static output feedback designs for
hyperbolic 2×2 systems
In the nonlinear case, local stabilization in H2
Interesting open problem: N×N systems (slugging flows in offshore oil rig risers)