analysis and control of a conservation law with …analysis and control of a conservation law with...

Introduction Main results Future work

Analysis and control of a conservation lawwith nonlocal velocity

Zhiqiang WANG

Fudan University, Shanghai, China

February 3, 2012Laboratoire Jacques-Louis Lions


Acknowledgements

Séminaire du Groupe de Travail ContrôleLaboratoire Jacques-Louis LionsFondation Sciences Mathématiques de Paris2009 Benasque Summer School2010 Triemestre IHP

Joint work with J.-M. Coron, M. Gröschel, M. Gugat, M. Kawski,A. Keimer, G. Leugering and P. Shang


Outline

1 IntroductionMotivationProblems

2 Main resultsAnalysis

Well-posedness of Forward SystemWell-posedness of Adjoint System

ControlExact (State) ControllabilityTime-optimal Transition Between EquilibriaNodal Profile ControllabilityStabilizationOptimal Control Problems

3 Future work


Motivation

Semiconductor wafer manufacture system


Motivation

Semiconductor wafer manufacture system

The semiconductor manufacturing system which has a highlyre-entrant character.

very high volume (number of parts manufactured per unittime) at each processvery large number of consecutive production steps

Cycle time: 6 weeks Work steps: 500 WIP: 60,000 wafers


Problems

Model

1-D hyperbolic system (conservation law with nonlocal velocity)[Armbruster et al., 2006]

ρt(t , x) + (ρ(t , x)λ(W (t)))x = 0, (t , x) ∈ (0, T )× (0, 1),

ρ(0, x) = ρ0(x), x ∈ (0, 1),

ρ(t , 0)λ(W (t)) = u(t), t ∈ (0, T ).

(1)with W (t) =

∫ 10 ρ(t , x)dx .

λ(·) ∈ C1([0,+∞); (0,+∞)) and is usually decreasing.A typical special case: λ(W ) = 1

1+W .


Problems

The space variable x ∈ [0, 1] represents the wholemanufacture process. x = 0 corresponds to the entry ofthe factory (manufacture line) while x = 1 corresponds tothe exit of the factory (manufacture line).ρ(t , x) ≥ 0 is the density of product at time t and stage x.ρ(t , ·) (t ≥ 0) is called the state in exact controllabilityproblem.The control u(t) of this system is acted on the influxρ(t , 0)λ(W (t)), the rate of products entering the factory.The natural output is the outflux y(t) := ρ(t , 1)λ(W (t)), therate of products exiting the factory.


Problems

Some control problems (I)

Exact (state) Controllability ProblemPossibility to drive arbitrary initial state ρ0 to arbitrary finalstate ρ1 over some [0, T ].Time-optimal Transition ProblemFind time-optimal control to drive given initial state ρ0 togiven final state ρ1.Stabilization ProblemPossibility to obtain asymptotic stability of the closed-loopsystem by constructing a feedback control. Naturalfeedback u(t) = ky(t). ‖ρ(t , ·)− ρ‖ → 0 as t →∞.


Problems

Some control problems (II)

Nodal Profile (Outflux) Controllability ProblemPossibility to drive the system, which starts from arbitraryinitial state ρ0, to reach exactly some given value yd for theoutflux y over [T1, T2].Demand Tracking ProblemMinimize the Error Signal ‖y − yd‖,the difference between the actual outflux y(t) and a givendemand forecast yd(t).Backlog ProblemMinimize the Accumulated Error Signal ‖β‖,β(t) =

∫ t0 y(s) ds −

∫ t0 yd(s) ds.


Problems

Related work

1 D. Armbruster, D. Marthaler, C. Ringhofer, K. Kempf andT.-C. Jo, A continuum model for a re-entrant factory, Oper.Res., 2006.

2 M. La Marca, D. Armbruster, M. Herty, and C. Ringhofer,Control of continuum models of production systems, IEEETrans. Automatic Control, 2010.

3 R. Colombo, M. Herty, and M. Mercier, Control of thecontinuity equation with a non local flow, ESAIM ControlOptim. Calc. Var., 2010.


Analysis

Definition of weak solution in L1

Let T > 0, ρ0 ∈ L1(0, 1) and u ∈ L1(0, T ) be given. A weaksolution of the system (1) is a function ρ ∈ C0([0, T ]; L1(0, 1))such that, for every τ ∈ [0, T ] and every ϕ ∈ C1([0, τ ]× [0, 1])such that

ϕ(τ, x) = 0,∀x ∈ [0, 1] and ϕ(t , 1) = 0,∀t ∈ [0, τ ],

one has ∫ τ

0

∫ 1

0ρ(t , x)(ϕt(t , x) + λ(W (t))ϕx(t , x))dxdt

+

∫ τ

0u(t)ϕ(t , 0)dt +

∫ 1

0ρ0(x)ϕ(0, x)dx = 0.


Analysis

Well-posedness in L1

Theorem

If ρ0 ∈ L1+(0, 1) and u ∈ L1

+(0, T ), then system (1) admits aunique weak solution ρ ∈ C0([0, T ]; L1

+(0, 1)), which is alsononnegative almost everywhere in Q = [0, T ]× [0, 1].

Proof sketch.1 Existence and uniqueness of local solution

Characteristic method + fixed point argument bycontraction mapping principle.Explicit expression of ρ in terms of ξ(t) :=

∫ t0 λ(W τ))dτ .

2 Existence of global solutionUniform a priori estimate of ‖ρ(t , ·)‖L1(0,1).


Analysis

Regularity

Remark (Hidden regularity)

ρ ∈ C0([0, T ]; L1(0, 1)) ⊂ L1(0, 1; L1(0, T )) ρ ∈ C0([0, 1]; L1(0, T )).

Remark (Lp Regularity)

ρ0 ∈ Lp+(0, 1) and u ∈ Lp

+(0, T ) (1 ≤ p < ∞)=⇒ ρ ∈ C0([0, T ]; Lp

+(0, 1)) ∩ C0([0, 1]; Lp+(0, T )).

Remark (W 1,p Regularity)

ρ0 ∈ W 1,p+ (0, 1) and u ∈ W 1,p

+ (0, T ) (1 ≤ p < ∞) withρ0(0) = u(0)

=⇒ ρ ∈ C0([0, T ]; W 1,p+ (0, 1)) ∩ C0([0, 1]; W 1,p

+ (0, T )).


Analysis

Adjoint system

Suppose that ρ ∈ C0([0, T ]; W 1,p+ (0, 1)) ∩ C0([0, 1]; W 1,p

+ (0, T ))is solution to (1). The adjoint system is defined as

qt(t , x) + (q(t , x)λ(W (t)))x

= −λ(W (t))∫ 1

0 ρ(t , s)qx(t , s)ds, (t , x) ∈ (0, T )× (0, 1),

q(T , x) = q0(x), x ∈ (0, 1),

q(t , 1) = v(t), t ∈ (0, T ).

(2)where W (t) =

∫ 10 ρ(t , x)dx .


Analysis

Well-posedness of adjoint system

Theorem

Let p ∈ [1,∞), q0 ∈ W 1,p(0, 1) and v ∈ W 1,p(0, T ) withq0(0) = v(0), then (2) has a unique solutionρ ∈ C0([0, T ], W 1,p(0, 1)) ∩ C0([0, 1], W 1,p(0, T )).


Analysis

Proof sketch (I)

Step 1It suffices to study

qt(t , x) + α(t)qx(t , x)

= −∫ 1

0 γ(t , s)qx(t , s)ds, (t , x) ∈ (0, T )× (0, 1),

q(0, x) = q0(x), x ∈ (0, 1),

q(t , 0) = v(t), t ∈ (0, T ).

(3)

where α ∈ C0([0, T ]; (0,∞)) and γ ∈ C0+([0, T ]× [0, 1]).


Analysis

Proof sketch (II)

Step 2Differentiate (3) with respect to x :

qx t(t , x) + α(t)qx x(t , x) = 0, (t , x) ∈ (0, T )× (0, 1),

qx(0, x) = q′0(x), x ∈ (0, 1),

qx(t , 0) == − 1α(t) ·

(v ′(t) +

∫ 10 γ(t , s)qx(t , s)ds

), t ∈ (0, T ).

(4)Let f (t) :=

∫ 10 γ(t , s)qx(t , s)ds.


Analysis

Proof sketch (III)

Step 3

f (t) = g(t) +

∫ t

0k(t , τ) f (τ)dτ, (5)

where k(t , τ) := −γ(

t ,∫ tτ α(r)dr

)and

g(t) :=−∫ t

0γ(

t ,∫ t

τα(r)dr

)v ′(τ)dτ

+

∫ 1−R t

0 α(r)dr

0γ(t , s +

∫ t

τα(r)dr)q′0(s)ds

are continuous.

(5) =⇒ f(4)

=⇒ qx

R x0=⇒ q.


Control

Definition of exact (state) controllability

Exact (State) Controllability

State ρ(t , ·) ∈ Lp+(0, 1).

Does there exists Control u(t) ∈ Lp+(0, T ) to drive the

hyperbolic system (1): ρ0 ∈ Lp+(0, 1) ρ1 ∈ Lp

+(0, 1) on [0, T ],i.e.,

ρ(T , x) = ρ1(x), x ∈ (0, 1).


Control

Classical result can not be applied

Li-Rao’s constructive method.Key idea: ut + A(u)ux = 0 ⇐⇒ ux + A−1(u)ut = 0 when A(u)has no zero eigenvalues.

Tatsien Li, Bopeng Rao, Exact boundary controllability forquasi-linear hyperbolic systems, SIAM J. Control Optim.,41(6):1748–1755 (electronic), 2003.Tatsien Li, Controllability and observability for quasilinearhyperbolic systems, volume 3 of AIMS Series on AppliedMathematics. American Institute of Mathematical Sciences(AIMS), Springfield, MO, 2010.


Control

Local controllability around ρ

Theorem

Let ρ ≥ 0 be the given constant equilibrium and T0 := 1λ(ρ) .

Then, for any T > T0, and any p ≥ 1, the system (1) is locallycontrollable.

Proof sketch.1 Controllability for linear control system

ξ =⇒ u =⇒ ρ

2 Fixed point argument by contraction mapping principleξ =⇒ u =⇒ ρ =⇒ F (ξ) = ξ


Control

Global controllability

Theorem

For any p ∈ [1,+∞), any ρ0 ∈ Lp+(0, 1) and any ρ1 ∈ Lp

+(0, 1),there exists T1 > 0 (depending on ρ0 and ρ1) such that for anyT ≥ T1, the system (1) is controllable.

Proof sketch.1 Drive the state from ρ0 at t = 0 to some equilibrium ρ at

t = Tf .Input control u(t) can be induced by natural state controlρ(t , 0) ≡ ρ.

2 Drive the system from ρ at t = Tf to ρ1 at t = T := Tf + Tbby using the reversibility of the hyperbolic system(t , x , ρ(t , x)) → (T − t , 1− x , ρ(T − t , 1− x)).


Control

Transition between equilibria

Exact controllabilityDrive the system from one equilibrium ρ0 to another ρ1.Admissible control set.

Time-optimal controlThe global controllability time is not optimal.What is the optimal control time to drive the system (1) fromone equilibrium state ρ0 to another equilibrium state ρ1.What is the control?


Control

An intuitive time-optimal control

Choose the intuitive density controlρt(t , x) + (ρ(t , x)λ(W (t)))x = 0, (t , x) ∈ (0, T )× (0, 1),

ρ(0, x) = ρ0, x ∈ (0, 1),

ρ(t , 0) = ρ1, t ∈ (0, T )

for the special case: λ(W ) = 11+W .

ρ(T , x) ≡ ρ1 for T ≥ 1 + ρ0+ρ12

u(t) = ρ1√(1+ρ0)2+2t(ρ1−ρ0)

, t ∈ (0, T ).


Control

Time-optimal control of transition between equilibria

TheoremThe minimum time to transfer the state from one equilibrium ρ0to the other equilibrium ρ1 using nonnegative influx controlu ∈ L1

+(0,∞) is T = 1 + ρ0+ρ12 . The time-optimal control is

indeed the natural one u(t) = ρ1√(1+ρ0)2+2t(ρ1−ρ0)

, t ∈ (0, T ).

Proof.Direct computations.


Control

Definition of nodal profile controllability

Where it comes?Nodal profile controllability was originally introduced by M.Gugat et al. for gas demanding control. This kind ofcontrollability was later named by T. Li and generalized for firstorder quasilinear hyperbolic systems.

DefinitionFor any given initial data ρ0, boundary data yd and any T1, Twith 0 < T1 < T , to find suitable suitable controlu : (0, T ) 7→ [0,+∞) such that the solution ρ to the system (1)satisfies also the nodal profile condition:

ρ(t , 1)λ(W (t)) = yd(t), t ∈ (T1, T ). (6)


Control

Nodal profile controllability

Theorem

Let ρ ≥ 0 be the given constant equilibrium and let T0 := 1λ(ρ) .

For any p ∈ [1,+∞), and any T1, T with T0 < T1 < T , thereexists ν > 0 such that the following holds: For any ρ0 ∈ Lp

+(0, 1)and any yd ∈ Lp

+(T1, T )

‖ρ0(·)− ρ‖Lp(0,1) ≤ ν, ‖y(·)− ρλ(ρ)‖Lp(T1,T ) ≤ ν,

there exists u ∈ Lp+(0, T ) such that the weak solution

ρ ∈ C0([0, T ]; Lp(0, 1)) to the system (1) satisfies the out-fluxcondition (6).

Proof sketch.Nodal profile controllability for linear systemFixed point argument.


Control

Output feedback stabilization

Definition of output feedback stabilization

ρt(t , x) + (ρ(t , x)λ(W (t)))x = 0, (t , x) ∈ (0,∞)× (0, 1),

ρ(0, x) = ρ0(x), x ∈ (0, 1),

u(t)− ρλ(ρ) = k(y(t)− ρλ(ρ)), t ∈ (0,∞).

(7)with W (t) =

∫ 10 ρ(t , x)dx , u(t) = ρ(t , 0)λ(W (t)), y(t) =

ρ(t , 1)λ(W (t)).

Can we find k such that ‖ρ− ρ‖ → 0 as t →∞?


Control

Simple observation

Does k = 0 work?u(t) = ρλ(ρ).

For ρ = 0, YES, it works!u(t) = 0 ⇐⇒ ρ(t , 0) = 0While for ρ 6= 0, NOT clear!


Control

Linearization near constant ρ ≥ 0

Linearized systemLet

d :=ρλ′(ρ)

λ(ρ). (8)

The difference function ρ := ρ− ρ. Omitting , then ρ satisfiesρt(t , x) + ρx(t , x) = 0, t ∈ (0,∞), x ∈ (0, 1),

ρ(0, x) = ρ0(x), x ∈ (0, 1),

ρ(t , 0) = kρ(t , 1) + (k − 1)dW (t), t ∈ (0,∞),

(9)

where W (t) =∫ 1

0 ρ(t , x)dx and

λ(ρ) = 1

without loss of generality.


Control

IFF condition for stabilization

Theorem

Let ρ ≥ 0. Then, ρ ∈ L2(0, 1) is exponentially stable in L2(0, 1)for the control system (9) if and only if d > −1 and |k | < 1.That is to say: if and only if d > −1 and |k | < 1, there existconstants C = C(d , k) > 0 and α = α(d , k) > 0 such that thefollowing holds: For any ρ0 ∈ L2(0, 1), the weak solutionρ ∈ C0([0,∞); L2(0, 1)) to the system (9) satisfies

‖ρ(t , ·)‖L2(0,1) ≤ Ce−αt · ‖ρ0‖L2(0,1), ∀t ∈ [0,∞). (10)


Control

Proof by spectral analysis

Characteristic equationLet µ ∈ C and φ 6= 0 be an eigen pair s.t.

µφ(x) + φ′(x) = 0, x ∈ (0, 1),

φ(0) = kφ(1) + (k − 1)d ·∫ 1

0φ(x)dx .

which impliesφ(x) = e−µx ,

1− ke−µ + (1− k)d ·∫ 1

0e−µxdx = 0.

(11)


Control

Spectral condition

Proposition(11) has no solution (µ, φ) such that <(µ) ≥ 0 and φ 6= 0 if andonly if d > −1 and |k | < 1.

Proof sketch.Degree theory homotopic functions

1 d = −1 and k ∈ R or d 6= −1 and k = 1. ∃µ = 02 d 6= −1 and |k | > 1. ∃<(µ) > 03 d < −1 and |k | < 1 or k = −1. ∃µ ∈ (0,∞)

4 d > −1 and k = −1. ∃µ = ib, b ∈ R, b 6= 05 d > −1 and |k | < 1. No <µ ≥ 0


Control

Lyapunov function approach

Case |d | < 1 and |k | < 1

Define

L(t) :=

∫ 1

0e−βxρ2(t , x)dx + aW 2(t), (12)

By letting β → 0+, a := e−β−k1−k d > 0,

L(t) ≤ − β

C[1− d2(eβ − 1)2e−ββ−2] · L(t) ≤ −αL(t),

since 1− d2(eβ − 1)2e−ββ−2 → 1− d2 > 0 as β → 0+.


Control


Case d ≥ 1 and |k | < 1

Define

V1(t) :=

∫ 1

0ρ2(t , x)dx + bW 2(t), (13)

Choose b = d , then

V1(t) = (k2 − 1)ξ2(t , 1) ≤ 0, ∀t ≥ 0,

whereξ(t , x) := ρ(t , x) + dW (t)

satisfiesξt(t , x) + ξx(t , x) = dW (t), t ∈ (0,∞), x ∈ (0, 1),

ξ(0, x) = ρ0(x)− ρ + dW (0), x ∈ (0, 1),

ξ(t , 0) = kξ(t , 1), t ∈ (0,∞).

(14)


Control


Case d ≥ 1 and |k | < 1

Define

V2(t) :=

∫ 1

0e−xξ2(t , x)dx , (15)

thenV2(t) ≤ −

12

V2(t) + Aξ2(t , 1).

LetV (t) :=

A1− k2 V1(t) + V2(t). (16)

Then

V (t) =A

1− k2 V1(t) + V2(t) ≤ −12

V2(t) ≤ −αV (t), ∀t ≥ 0.


Control

Stabilization to ρ = 0 for nonlinear system

ρ = 0 =⇒ d = 0.

TheoremFor any k ∈ (−1, 1) and any R > 0, there exist constantsC = C(k , R) > 0 and α = α(k , R) such that for anyρ0 ∈ L2((0, 1); [0,∞)) with

‖ρ0‖L1(0,1) ≤ R, (17)

the solution ρ ∈ C0([0,∞); L2(0, 1)) to the system with feedback

u(t) = ky(t)

satisfies

‖ρ(t , ·)‖L2(0,1) ≤ Ce−αt · ‖ρ0‖L2(0,1), ∀t ∈ [0,∞). (18)


Control

Proof by Lyapunov approach

Proof.Define

L(t) :=

∫ 1

0e−βx |ρ(t , x)|2dx , ∀t ∈ [0,∞),

then

L(t) =− βλ(W (t))L(t) + (λ(W (t)))−1(k2 − e−β)y2(t),≤ −αL(t), t ∈ (0,∞)

by β → 0+ and

0 ≤ W (t) ≤ ‖ρ0‖L1(0,1) ≤ R, ∀t ∈ [0,∞).


Control

Stabilization to ρ > 0 for nonlinear system

TheoremLet d > −1. For any k ∈ (−1, 1), there exist constantsε = ε(d , k) > 0, C = C(d , k) > 0 and α = α(d , k) > 0 such thatthe following holds: For any ρ0 ∈ L2((0, 1); [0,∞)) with

‖ρ0(·)− ρ‖L2(0,1) ≤ ε,

the weak solution ρ ∈ C0([0,∞); L2(0, 1)) to the system with

u(t)− ρλ(ρ) = k(y(t)− ρλ(ρ)), t ∈ (0,∞)

satisfies

‖ρ(t , ·)− ρ‖L2(0,1) ≤ Ce−αt · ‖ρ0(·)− ρ‖L2(0,1), ∀t ∈ [0,∞).


Control


Case d ∈ (−1, 1) and |k | < 1 .

Define L(t) as (12), then

L(t) ≤ − β

C2

(1− d2(eβ − 1)2e−ββ−2 + O(ε)

)· L(t)

≤ −αL(t),

letting first β = β(d , k) → 0+ and then ε = ε(d , k) → 0+, since1− d2(eβ − 1)2e−ββ−2 → 1− d2 > 0 as β → 0+.


Control


Case d ≥ 1 and |k | < 1 .

Define V1(t), V2(t), V (t) as (13),(15),(16), then

V (t) ≤ (−12

+ O(ε))V2(t) ≤ −αV (t), ∀t ≥ 0

by letting ε = ε(d , k) → 0+.


Control

Optimal Control Problem (I)

Optimal Control Problem without constrains

For any fixed Demand Forecast yd ∈ L2+(0, T ) and initial data

ρ0 ∈ L2+(0, 1), we look for solution to the optimal control

problem:

minu∈L2

+(0,T )J(u) := ‖u‖2

L2(0,T ) + ‖y − yd‖2L2(0,T ),

where y(t) = ρ(t , 1)λ(W (t)) is the outflux corresponding to theinflux u ∈ L2

+(0, T ) and initial data ρ0 ∈ L2+(0, 1).


Control

Solution to optimal control problem

Theorem

The infimum of the functional J(u) in L2+(0, T ) is achieved, i.e.,

there exists u∞ ∈ L2+(0, T ) such that J(u∞) = infu∈L2

+(0,T ) J(u).

Proof sketch.1 Choose a minimizing sequence un∞n=1.

Take a subsequence un u∞ weakly in L2+(0, T ).

2 Solve system (1) corresponding to influx un (resp., u∞) toobtain solution ρn (resp., ρ∞).Prove that yn(·) := λ(Wn(·))ρn(·, 1) y∞(·):= λ(W∞(·))ρ∞(·, 1) weakly in L2(0, T ).


Control

Optimal Control Problem (II)

Optimal control problem with constrains

minu∈U,y∈Y

J(u, y)

subject toρt(t , x) + (ρ(t , x)λ(W (t)))x = 0, (t , x) ∈ (0, T )× (0, 1),

ρ(0, x) = ρ0(x), x ∈ (0, 1),

ρ(t , 0)λ(W (t)) = u(t), t ∈ (0, T ),

ρ(t , 1)λ(W (t)) = y(t), t ∈ (0, T ),

with W (t) =∫ 1

0 ρ(t , x)dx .


Control

Basic setting of cost functional J

Basic setting of J

For given U ⊆ L2+(0, T ) and Y ⊆ L2

+(0, T ), J : U × Y → R isassumed to be Fréchet differentiable in L2(0, T ) and∂J(u,y)

∂u , ∂J(u,y)∂y ∈ H1(0, T ).

Example

U = Y ⊆ H1+(0, T )

J(u, y) = κ2‖u − ud‖2

L2(0,T )+ ν

2‖y − yd‖2L2(0,T )

, with

ud , yd ∈ H1(0, T ) and κ, ν ∈ (0,∞).∂J(u,y)

∂u = κ(u − ud), ∂J(u,y)∂y = ν(y − yd).


Control

Lagrange approach

Definition of Lagrange functional

L : (u, y , ρ, q) ∈ U × Y × H1(ΩT )× H1(ΩT ) 7→ R

with

L(u, y , ρ, q) :=J(u, y)− < qt + λ(W (·))qx , ρ >L2(ΩT )

+ < q(T , ·), ρ(T , ·) >L2(0,1) − < q(0, ·), ρ0 >L2(0,1)

+ < q(·, 1), y >L2(0,T ) − < q(·, 0), u >L2(0,T ) .


Control

KKT conditions

TheoremThe minimizer (u, y) ∈ U × Yof J satisfies problems

ρt(t , x) + (ρ(t , x)λ(W (t)))x = 0, (t , x) ∈ (0, T )× (0, 1),

ρ(0, x) = ρ0(x), x ∈ (0, 1),

ρ(t , 0)λ(W (t)) = u(t), t ∈ (0, T ),

ρ(t , 1)λ(W (t)) = y(t), t ∈ (0, T ),

qt(t , x) + (q(t , x)λ(W (t)))x

= −λ(W (t))∫ 1

0 ρ(t , s)qx(t , s)ds, (t , x) ∈ (0, T )× (0, 1),

q(T , x) = 0, x ∈ (0, 1),

q(t , 1) = −∂J(u,y)∂y , t ∈ (0, T )

q(t , 0) = ∂J(u,y)∂u , t ∈ (0, T ).


Control

KKT conditions tell what?

How to find optimal control

u(1)

=⇒ ρ =⇒ W , y(2)

=⇒ q =⇒ q(t , 0) =∂J∂u

(u∗, y) =⇒ u∗

Example

U = Y ⊆ H1+(0, T )

J(u, y) = κ2‖u − ud‖2

L2(0,T )+ ν

2‖y − yd‖2L2(0,T )

, with

ud , yd ∈ H1(0, T ) and κ, ν ∈ (0,∞).

∂J∂u

(u∗, y) = κ(u∗ − ud) =⇒ u∗ = ud +1κ

q(t , 0)


Our articles related to this work

J.-M. Coron, M. Kawski, Z. Wang, Analysis of aconservation law modeling a highly re-entrantmanufacturing system, Discrete Contin. Dyn. Syst. Ser. B,2010.P. Shang, Z. Wang, Analysis and control of a scalarconservation law modeling a highly re-entrantmanufacturing system, J. Differential Equations, 2011.J.-M. Coron, Z. Wang, Controllability for a scalarconservation law with nonlocal velocity, J. DifferentialEquations, 2012.


Work in progress

J.-M. Coron, Z. Wang, Output feedback stabilization for ascalar conservation law with nonlocal velocity, in prepare.M. Gröschel, A. Keimer, G. Leugering, Z. Wang, Regularitytheory and adjoint based optimality conditions for anonlinear transport equation with nonlocal velocity, inprepare.M. Gugat, A. Keimer, Z. Wang, Optimal control of anetwork of re-entrant factories with input and capacityconstraints, in prepare.


Future work

Other applicationParticle synthesis process, Follicle ovulationGeneralizationNetworks, Coupled systems, Higher dimensionNumerical experimentsStabilization with other types of feedbackOptimal control problems with constrains


Thank you for your attention!

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GBKkai

analysis and control of a conservation law with …analysis and control of a conservation law with...

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