[PDF] Coopération ville-hôpital: évaluation de la faisabilité et de lintérêt de

View - Download Coopération ville-hôpital: évaluation de la faisabilité et de lintérêt de

Previous PDF

Next PDF

>G A/, i2H@yRNy9399 ?iiTb,ffTbi2HX?HXb+B2M+2fi2H@yRNy9399 _Q#mbi /2bB;M Q7 #+Fbi2TTBM; +QMi`QHH2`b 7Q` bvbi2Kb Q7

HBM2`?vT2`#QHB+ S.1b

hQ +Bi2 i?Bb p2`bBQM,

THÈSE DE DOCTORAT

de l"Université de recherche Paris Sciences et Lettres

PSL Research University

Préparée à MINES ParisTechContrôle robuste d"EDPs linéaires hyperboliques par méthodes

de backsteppingÉcole doctorale n o432

SCIENCES ET MÉTIERS DE L"INGÉNIEURSpécialitéMATHÉMATIQUES ET AUTOMATIQUESoutenue parJean AURIOL

le 4 Juillet 2018

Dirigée parFlorent DI MEGLIO

Nicolas PETITCOMPOSITION DU JURY :

M. Yann Le Gorrec

FEMTO-ST, Président

M. Alexey Pavlov

Norwegian University of Science

and Technology, Rapporteur

M. Rafael Vázquez

Universidad de Sevilla, Rapporteur

Mme. Kirsten Morris

University of Waterloo, Examinateur

M. Silviu Iulian Niculescu

L2S, Examinateur

M. Nicolas Petit

MINES ParisTech, Examinateur

M. Florent Di Meglio

MINES ParisTech, Examinateur

Remerciements

Contents

1 Introduction5

I Control theory 17

2 A primer on boundary controllability and observability for linear first order

hyperbolic systems 19

3 One-sided boundary stabilization and observability 41

4 Two-sided boundary stabilization and observability 57

(n+m)

5 An explicit mapping from LFOH PDEs to neutral systems 93

n+m

II Robust stabilization 111

6 Delay-robust stabilization 115

n+m Contents7 Delay-robust stabilization of a hyperbolic PDE-ODE system 131

8 Disturbance rejection and Input-to-State Stability (ISS) for a system of two

equations145

Conclusions and perspectives 179

Bibliography181

A Late-lumping approximation 193

B Convexity properties of the robustness domain 203

Chapter 1

Introduction

Context

Chapter 1. Introduction

n+1 n strong stability + uncoupled

Problems addressed and thesis organization

n+m equal n+m

Chapter 1. Introduction

Contribution 1:

Contribution 2:

Contribution 3:

Publications

minimum-time control of heterodirectional linear coupled hyperbolic PDEs Two sided boundary stabilization of heterodirectional linear coupled hyperbolic PDEs

Delay-robust control design for

two heterodirectional linear coupled hyperbolic PDEs

Delay-

robust stabilization of a hyperbolic PDE-ODE system

Late-lumping backstepping control of partial

differential equations An explicit mapping from linear first order hyperbolic PDEs to difference systems A sufficient stability condition for a class of linear 2 x 2 hyperbolic PDEs using a back- stepping transform Two sided boundary stabilization of two linear hyperbolic

PDEs in minimum-time

Trajectory tracking for a system of two linear hyperbolic PDEs with uncertainties

Robust output regulation of

2×2 hyperbolic systems: Control law and Input-to-State Stability

Introduction

Chapter 1. Introduction

n+ 1 n stabilité forte + Problèmes considérés et organisation de la thèse n+m

égales

n+m

Chapter 1. Introduction

Contribution 1:

Contribution 2:

Contribution 3:

Publications

minimum-time control of heterodirectional linear coupled hyperbolic PDEs Two sided boundary stabilization of heterodirectional linear coupled hyperbolic PDEs

Delay-robust control design for

two heterodirectional linear coupled hyperbolic PDEs

Delay-

robust stabilization of a hyperbolic PDE-ODE system

Late-lumping backstepping control of partial

differential equations An explicit mapping from linear first order hyperbolic PDEs to difference systems A sufficient stability condition for a class of linear 2 x 2 hyperbolic PDEs using a back- stepping transform Two sided boundary stabilization of two linear hyperbolic

PDEs in minimum-time

Trajectory tracking for a system of two linear hyperbolic PDEs with uncertainties

Robust output regulation of

2×2 hyperbolic systems: Control law and Input-to-State Stability

Notations

p?N? X?Rp X ||X||=? ???p i=1(Xi)2. p?N?q?N? Mp,q(R) p×q q IdqId p?N?q?N? K? Mp,q(R) ||K||= max{||Kξ||;ξ?Rp,||x||= 1}. p?N? K? Mp,p (K) K n?N (p1,···,pn)?Rn D=diag{p1,···,pn}

Cq([0,1])q?N? {∞} [0,1] q

qth C0([0,1])

C([0,1])

L1([0,1],R) L1([0,1])

[0,1]

L1 f?L1([0,1])

?f?L1=? 1

0|f(x)|dx.

L2([0,1],R) [0,1]

L2 f?L2([0,1],R)

||f||L2=?? 1

0f2(x)dx.

L∞([0,1],R) [0,1]

L∞i.e. f?L∞([0,1],R)

||f||L∞= sup x?[0,1]|f(x)|. Chapter 1. Introduction H1([0,1],R) =W1,2([0,1],R) f

L2([0,1],R) f 1 L2

H1 f?H1([0,1],R)

||f||H1=?? 1

0f2(x)dx+?

1

0(f?(x))2dx.

Ω?R

1

Ω(θ) =?1 ifθ?Ω

0 otherwise.

k1 a < b f [a,b] k

1 (x,y)?[a,b]2

s?C ?(s)

C+ C+={s?C,?(s)≥0}

ˆf(s) f(t)

(u,v)?(L2([0,1]))2 X?Rp ||(u,v,X)||2=||u||L2+||v||L2+||X||

A g(·)

A g(t) =gr(t) +∞? i=0g iδ(t-ti), gr?L1(R+,R)? i≥0|gi|<∞0 =t0< t1< ...δ ?g?A=?gr?L1+? i≥0|gi|.

ˆA A

?ˆg?ˆA=?g?A

K h:R+→R+h(0) = 0

KL g:R+×R+→R+ t≥0

g(·,t) K s≥0 g(s,·) i,j p Ep i,j? Mp,p(R) ith jth

Part I

Control theory

Chapter 2

A primer on boundary

controllability and observability for linear first order hyperbolic systems Chapitre 2: Commandabilité et observabilié frontière pour des systèmes hyper- boliques linéaires du premier ordre. backstepping Contents2.1 System under consideration and well-posedness . . . . . . . . . . . . . .20

2.2 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

2.3 Boundary controllability and observability problems . . . . . . . . . . .

27

2.4 One-sided controllabilty and observability: tutorial case of two equations

31

2.5 Summary and organization of Part I . . . . . . . . . . . . . . . . . . . . .

38
Chapter 2. A primer on boundary controllability and observability for linear first order hyperbolic systems

2.1 System under consideration and well-posedness

tu(t,x) + Λ+∂xu(t,x) = Σ++(x)u(t,x) + Σ+-(x)v(t,x), tv(t,x)-Λ-∂xv(t,x) = Σ-+(x)u(t,x) + Σ--(x)v(t,x), u(t,0) =Q0v(t,0) +U(t), v(t,1) =R1u(t,1) +V(t), u= (u1...un)T, v= (v1...vm)T (t,x) {(t,x)|0< t < T, x?[0,1]} RnRm Λ+Λ- n×nm×m 10

0λn)

10

0μm)

-μm<···<-μ1<0< λ1<···< λn.

C0([0,1],R) Q0

R1 n×mm×n

UV RnRm

?u 0 v 0? (u1)0(·)...(un)0(·) (v1)0(·)...(vm)0(·)? T, (L2([0,1]))n+m Remark 2.1.1Even if the matricesΛ+andΛ-are assumed to be constant in this thesis, most of the presented results can be extended to the case of spatially varying matrices whose components areC1([0,1],R)-functions.

2.1. System under consideration and well-posedness

Definition 2.1.1. T >0 ?

u 0v0? ?(L2([0,1]))n+mU? (L2([0,T]))nV?(L2([0,T]))m (u,v) (u,v)?(C0([0,T];L2([0,1])))n+m 0 = 0? 1

0(-∂tφT(t,x)-∂xφT(t,x)Λ+-φT(t,x)Σ++(x)-ψT(t,x)Σ-+(x))u(t,x) + (-∂tψT(t,x)

1

0φT(τ,x)u(τ,x)

-φT(0,x)u(0,x) +ψT(τ,x)v(τ,x)-ψT(0,x)v(0,x)dx-?

0φT(t,0)Λ+U(t) +ψT(t,1)Λ-V(t)dt

0(φT(t,1)Λ+-ψT(t,1)Λ-R1)u(t,1) + (ψT(t,0)Λ--φT(t,0)Λ+Q0)v(t,0)dt,

(φ,ψ)? C1([0,τ]×[0,1])n+m φ(·,1) =ψ(·,0) = 0 τ?[0,T] ddt u v? =A?u v? +B?U V? AB A

A:D(A)?(L2(0,1))n+m→(L2(0,1))n+m

?u v? ?-→?-Λ+∂xu+ Σ++(x)u+ Σ+-(x)v -∂xv+ Σ-+(x)u+ Σ--(x)v? D(A) ={(u,v)?(L2(0,1))n+m|u(0) =Q0v(0), v(1) =R1u(1)}.

A A?

A ?:D(A?)?(L2(0,1))n+m→(L2(0,1))n+m ?u v? ?-→?Λ+∂xu+ (Σ++(x))Tu+ (Σ-+(x))Tv -Λ-∂xv+ (Σ+-(x))Tu+ (Σ++(x))Tv? D(A?) ={(u,v)?(L2(0,1))n+m|u(1) = (Λ+)-1RT1Λ-v(t,1), v(0) = (Λ-)-1QT0Λ+u(t,0)}.

A C0(S(t))t≥0

Chapter 2. A primer on boundary controllability and observability for linear first order hyperbolic systemsDefinition 2.1.2.

Z A:D(A)? Z → Z

Theorem 2.1.1. [LP61, Corrolary 3.1]A D(A)

A ω <0

A? D(A?)

||z||D(A?)= (||z||2L2+||A?z||2L2) z?D(A?)

D(A?) ||·||D(A?)||·||H1 B? L(?n+m,D(A?)?)

< B ?U V? ,?z 1 z 2? >=z1(0)TΛ+U+z2(1)TΛ-V,

B?? L(D(A?),?n+m) B??z

1 z 2? =?z

1(0)TΛ+

z

2(1)TΛ-?

Bquotesdbs_dbs46.pdfusesText_46

[PDF] Le retour de l actionnaire

[PDF] le retour de la momie

[PDF] le retour film

[PDF] le rétrocontrole négatif de la testostérone

[PDF] Le Rêve

[PDF] le rêve du jaguar

[PDF] Le Rêve, Emile Zola

[PDF] le revenu des ménages

[PDF] le rhinocéros blanc

[PDF] le rhinoceros d'or livre

[PDF] le rhinocéros d'or pdf

[PDF] Le rideau de fer de Winston Churchill

[PDF] LE RIRE

[PDF] Le risque du crédit : exemple Crise Grecque

[PDF] Le risque infectieux SVT 3eme