Well-Posedness and Regularity of Partial Differential...

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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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State variable: satisfies

State space:

Control and output space:

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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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Write in form:

where generates a semigroup on

Facts: Not

Not

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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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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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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


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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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.


PDEs Abstract Setting

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8. Summary

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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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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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8. Summary

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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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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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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:



is much smoother!

second order system:

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Theorem 1. The abstract second order system is well-posed IFF


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!


Remark:

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8. Summary

Abstract Setting PDEs

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Multi-dimensional wave equation with Dirichlet control:

where

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It can be formulated in the form of abstract second order

system:


where

Lax-Milgram Theorem

Dirichlet map

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State 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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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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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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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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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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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


It can be formulated in the form of abstract second order

system:

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State space:


Theorem 5: Plate system is well-posed (geometry approach):



Remark: “ ” is not easy even for one-dimensional beam:

Laganese (1991) proved it through considering LQ problem:

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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


[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):


when

Proof: In the frequency domain, it is equivalent to proving

satisfies

Guess “ ”: By the one-dimensional case.

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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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8. Summary

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Mechanical formulation of isotropic linear elasticity:


Strain tensor:

Stress tensor:

where

Kronecker delta

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Another formulation of system of isotropic linear elasticity:


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:


where

State space: Control and output space:

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Theorem 7: The system is well-posed (PDE approach):



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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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.


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


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







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


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8. Summary

Wave equationWell-posedRegular SystemRegular System

Plate equation

Linear elasticity

Transmission plate equation

Shell equation


Schrödinger equation

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8. Summary

Wave equationWell-posedRegular SystemRegular System

Plate equation

Linear elasticity

Transmission plate equation

Shell equation


More need to be verified…

Schrödinger equation

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