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Academy of Mathematics and Systems Science Academia Sinica
Bao-Zhu Guo
Well-Posedness and Regularity of Partial Differential Equation Control Systems
29th Chinese Control Conference, Beijing, 2010
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Students and postdoctoral fellows
Z.X. ZhangZ.C. Shao
S.G. Chai
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Contents1. Motivation: String equation with boundary control
2. Background: Well-posed system
3. Background: Regular system
4. Abstract second order system
5. Multi-dimensional wave equation: Dirichlet boundary control
6. Multi-dimensional Euler-Bernoulli plate equation:Neumann boundary control
7. Linear elasticity system: Dirichlet boundary control
8. Summary
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1. Motivation: String equation with boundary control
Energy of the system:
Tension
Fig. Vibrating string
Elastic energy Kinetic energy
String
Vertical force
Output feedback stabilization of string equation with boundary pointwise control and observation:
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1. Motivation: String equation with boundary control
State variable: satisfies
State space:
Control and output space:
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1. Motivation: String equation with boundary control
Physically, everything is fine:
Input: Basic requirement for control and output
Output: Vertical force at
Energy
Theoretically, there are some problems!
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1. Motivation: String equation with boundary control
Write in form:
where generates a semigroup on
Facts: Not
Not
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1. Motivation: String equation with boundary control
Formally,
where is the extension of generates a semigroup on
Question 1: where is the solution for
Ideally, we can show by PDE approach that
is admissible!
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1. Motivation: String equation with boundary control
Question 2: where is the output?
Can we write for
Because for No more regularity!
Ideally, we can show by PDE approach that
is admissible!Answer: At least (Hidden regularity in PDE!)
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1. Motivation: String equation with boundary control
Actually, even if there is control, we can show by PDE approach that:
In any case:
Well-posed
+admissibility of
the output makes sense in the sense
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To sum up: There are some systems that
(a). Solution:Control operator is unbounded, but is not unbounded enough so that the trajectory goes out
(b). Output:Output operator is unbounded, but it is not unbounded enough so that the output goes out space!
Conclusion: It is significant to study these systems in the uniform abstract framework!
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Admissibility tells us that why we do not consider system in
1. Motivation: String equation with boundary control
Reason 1: is too large that system is not controllable in but it is in
Reason 2: may not lie in space for some
Necessity!
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1. Motivation: String equation with boundary control
Original system Exponential stable system:
Application of the FACT:
is the basic requirement for solvability of observer:
Input for observer!Control design:
Ref: [B.Z. Guo and C.Z.Xu, IEEE TAC(2007)]
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(distributed measurement).
J.Song, J.Y.Yu (1979): Considered the stabilization in
L.F.Ho, D.L.Russell (1983): realized that the energy space is the “optimal” space for such a system!
D.Salamon (1987): Abstract well-posed system (Bellman’s axiomatic theory for finite linear systems).
G.Weiss (1989): Abstract regular system.
1. Motivation: String equation with boundary control
PDEs Abstract Setting
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Contents1. Motivation: String equation with boundary control
2. Background: Well-posed system
3. Background: Regular system
4. Abstract second order system
5. Multi-dimensional wave equation: Dirichlet boundary control
6. Multi-dimensional Euler-Bernoulli plate equation:Neumann boundary control
7. Linear elasticity system: Dirichlet boundary control
8. Summary
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2. Background: Well-posed system
Definition (well-posed): Suppose thatThe system
is said to be Well-posed with state space , control space output space IF
1.
is admissible:
generator of a semigroup on
2.
3. is admissible:
4. Input-Output stable:
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2. Background: Well-posed system
Representation of a well-posed system
Time domain:
State:
Output:
Frequency domain:
NOT
Lebesgue extension of
can only determine uniquely,
Transfer function!
Drawbacks: is too complicated! Do not know
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Contents1. Motivation: String equation with boundary control
2. Background: Well-posed system
3. Background: Regular system
4. Abstract second order system
5. Multi-dimensional wave equation: Dirichlet boundary control
6. Multi-dimensional Euler-Bernoulli plate equation:Neumann boundary control
7. Linear elasticity system: Dirichlet boundary control
8. Summary
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3. Background: Regular system
Definition (regular): A well-posed system is called regular if zero is the Lebesgue point of the step response:
G.Weiss (1994)
Why regular?
Representation of regular system in time domain:
Differential equation:
Time domain:
Frequency domain:
Same form with finite LTI !
Feedthroughoperator!
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3. Background: Regular system
In frequency domain:
Roughly speaking:
Regular system is a class of infinite-dimensional systems that parallel in many ways to finite LTI ones!
Covers many PDEs with boundary control and measurement.
Theory has been fruitful (feedback, realization, LQ, etc….)
Question: What PDEs are well-posed or regular?
Same form with finite LTI !
Our work!
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Contents1. Motivation: String equation with boundary control
2. Background: Well-posed system
3. Background: Regular system
4. Abstract second order system
5. Multi-dimensional wave equation: Dirichlet boundary control
6. Multi-dimensional Euler-Bernoulli plate equation:Neumann boundary control
7. Linear elasticity system: Dirichlet boundary control
8. Summary
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Let be three Hilbert spaces. Consider the abstract
where
(i). (unbounded) positive self-adjoint;
(ii).
is given by (iii).
is bounded;
State space:
Control and output space:
4. Abstract second order system
is much smoother!
second order system:
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Theorem 1. The abstract second order system is well-posed IFF
4. Abstract second order system
Realized by R.Triggiani, a simple proof was given by [Guo(2009)].
Theorem 2. Well-posed
Exact controllable Exponential stable by feedback
Ref: [B.Z. Guo and Y.H. Luo , Systems & Control Letters(2002)].
are admissible!
Proposition: Transfer function
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Russell’s principle: For time invertible system (1973):
Exact controllable Exponential stable
Exponential stableExact controllable + Well-posed
The inverse is much difficult!
4. Abstract second order system
Remark:
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Contents1. Motivation: String equation with boundary control
2. Background: Well-posed system
3. Background: Regular system
4. Abstract second order system
5. Multi-dimensional wave equation: Dirichlet boundary control
6. Multi-dimensional Euler-Bernoulli plate equation:Neumann boundary control
7. Linear elasticity system: Dirichlet boundary control
8. Summary
Abstract Setting PDEs
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5. Multi-dimensional wave equation: Dirichlet boundary control
Multi-dimensional wave equation with Dirichlet control:
where
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It can be formulated in the form of abstract second order
system:
5. Multi-dimensional wave equation: Dirichlet boundary control
where
Lax-Milgram Theorem
Dirichlet map
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5. Multi-dimensional wave equation: Dirichlet boundary control
State space:
Control and output space:
Theorem 3: System is well-posed (PDE approach):
Corollary: Exact controllable Exponential stable by
P.F.Yao (1999) Y.X.Guo,S.G.Feng (2001)
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5. Multi-dimensional wave equation: Dirichlet boundary control
Theorem 4 (regularity): Wave system is regular as well andthe feedthrough operator (geometry approach):
where
Key point of proof: In frequency domain,
satisfies
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5. Multi-dimensional wave equation: Dirichlet boundary control
Introduce the inner product in tangent space
becomes a Riemannian manifold with Riemannian metric
Similar to constant case but work on manifold, we got the regularity in:
[B.Z. Guo and Z.X. Zhang, ESAIM(2007)]
Beltrami-Laplace operator
Regularity for variable coefficients: Riemannian geometry approach
Low order
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5. Multi-dimensional wave equation: Dirichlet boundary control
Exact controllable: Ho(1986), Lions(1988), Triggiani (1988)
Triggiani: “Exact boundary controllability… ”, AOM(1988)
Lasiecka and Triggiani: “Uniform exponential … ”, JDE (1987)
Well-posedness: Ammari (2002), yes;
Lasiecka and Triggiani (2003), No;
Lasiecka and Triggiani (2004), yes.
Remark on constant coefficients:
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5. Multi-dimensional wave equation: Dirichlet boundary control
In the polar coordinate:first kind Besselfunction
[B.Z.Guo and X.Zhang: SICON(2005)]
“Guess D”:
For dimension: Guo’s guess proved in:
Regularity for constant coefficients: PDE approach
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Contents1. Motivation: String equation with boundary control
2. Background: Well-posed system
3. Background: Regular system
4. Abstract second order system
5. Multi-dimensional wave equation: Dirichlet boundary control
6. Multi-dimensional Euler-Bernoulli plate equation:Neumann boundary control
7. Linear elasticity system: Dirichlet boundary control
8. Summary
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The multi-dimensional Euler-Bernoulli plate with Neumann boundary control and observation:
6. Multi-dimensional Euler-Bernoulli plate equation: Neumann boundary control
where
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where
6. Multi-dimensional Euler-Bernoulli plate equation: Neumann boundary control
It can be formulated in the form of abstract second order
system:
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State space:
6. Multi-dimensional Euler-Bernoulli plate equation: Neumann boundary control
Theorem 5: Plate system is well-posed (geometry approach):
Control and output space:
Corollary: Exact controllable Exponential stable by
Remark: “ ” is not easy even for one-dimensional beam:
Laganese (1991) proved it through considering LQ problem:
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6. Multi-dimensional Euler-Bernoulli plate equation: Neumann boundary control
a). Exact controllable: J.L.Lions, “Controllabilite exacte…”, (1988,French);
b). Exponential stable: N.Qurada and R.Triggiani, “Uniform stabilization… ”, DIE (1991).
d). Exact controllable: P. F. Yao, “Observability … ”, Contemporary Mathematics (2000).
c). Well-posedness: Collected from existing literature, in particular, I.Lasiecka and R.Triggiani (2003) .
e). Exponential stable: Our result’s consequence (much difficult in PDE).
Remark: Exact controllability in constant coefficients
Variable coefficients: Exact controllability:
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Key point of proof on well-posedness: On Riemannian
manifold that we defined previously
6. Multi-dimensional Euler-Bernoulli plate equation: Neumann boundary control
[B.Z. Guo and Z.X. Zhang, MCSS(2007)]
Similar to the constant case as Lasiecka and Triggiani(2003) butin Riemannian manifold to get the well-posedness in:
Levi-Civita connection with respect to
Beltrami-Laplace operator
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Theorem 6: The plate system is regular as well (geometry approach):
6. Multi-dimensional Euler-Bernoulli plate equation: Neumann boundary control
when
Proof: In the frequency domain, it is equivalent to proving
satisfies
Guess “ ”: By the one-dimensional case.
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6. Multi-dimensional Euler-Bernoulli plate equation: Neumann boundary control
Regularity with constant coefficients (PDE approach)was first established in:
[B.Z.Guo and Z.C.Shao, JDCS(2006)]
by flatting
Using Riemannian geometry method, the regularity with variable coefficients was established in:
[B.Z. Guo and Z.X. Zhang, MCSS(2007)]
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Contents1. Motivation: String equation with boundary control
2. Background: Well-posed system
3. Background: Regular system
4. Abstract second order system
5. Multi-dimensional wave equation: Dirichlet boundary control
6. Multi-dimensional Euler-Bernoulli plate equation:Neumann boundary control
7. Linear elasticity system: Dirichlet boundary control
8. Summary
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Mechanical formulation of isotropic linear elasticity:
7. Linear elasticity system: Dirichlet boundary control
Strain tensor:
Stress tensor:
where
Kronecker delta
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Another formulation of system of isotropic linear elasticity:
7. Linear elasticity system: Dirichlet boundary control
Linear elasticity is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions.
Strongly coupled system of wave equations!
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System can be transformed into the second order system:
7. Linear elasticity system: Dirichlet boundary control
where
State space: Control and output space:
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Theorem 7: The system is well-posed (PDE approach):
7. Linear elasticity system: Dirichlet boundary control
Corollary: Exact controllable Exponential stable by
Exact controllable: J.L.Lions (1988), V. Komornik (1998).
Strong stable: W. J. Liu and M. Krstic (2000).
Exponential stability is the consequence of Theorem 7!
Ref: [B.Z. Guo and Z.X. Zhang, SICON(2009)]
Open question: Exponential stable?
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7. Linear elasticity system: Dirichlet boundary control
Theorem 8: Linear elasticity system is regular as well and the feedthrough operator (geometry approach):
Remark: when , reduces to “n” independent wave equations:
Coincides with our first result for single wave equation in:
[B.Z.Guo and X.Zhang, SICON(2005)]!
Regularity: Big challenge due to strong coupling
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Key point of proof on regularity: Riemannian geometry method.
7. Linear elasticity system: Dirichlet boundary control
is viewed as a vector field on
The geometric formulation of linear elasticity on Riemannian
manifold that we had defined previously:
Speciality: Constant coefficients, but we still need geometry!
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where
7. Linear elasticity system: Dirichlet boundary control
exterior differentialand its formal adjoint
In frequency domain: we need to show that the solution
Ref: [S.G. Chai and B.Z. Guo, SICON(2010)]
which is the equation on Riemannian manifold satisfies
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Contents1. Motivation: String vibration control
2. Background: Well-posed system
3. Background: Regular system
4. Abstract second order system
5. Multi-dimensional wave equation: Dirichlet boundary control
6. Multi-dimensional Euler-Bernoulli plate equation:Neumann boundary control
7. Linear elasticity system: Dirichlet boundary control
8. Summary
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8. Summary
Well-posed system is a large class of infinite-dimensional systems covering those systems with boundary control and observation.
Most of well-posed systems are regular.
Regular systems parallel in many ways to finite-dimensional ones.
Abstract theory has been fruitful.
More PDEs are needed to be verified.
Regularity (well-posedness as well) is first appeared in PDEs.
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8. Summary
Well-posedRegular SystemRegular System
Well decorated! Not well furnished?
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8. Summary
Well-posedRegular SystemRegular System
Well decorated! Not well furnished?
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8. Summary
Wave equationWell-posedRegular SystemRegular System
Plate equation
Linear elasticity
Transmission plate equation
Shell equation
Well decorated! Not well furnished?
Schrödinger equation
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8. Summary
Wave equationWell-posedRegular SystemRegular System
Plate equation
Linear elasticity
Transmission plate equation
Shell equation
Well decorated! Not well furnished?
More need to be verified…
Schrödinger equation
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