Control theory for partial differential equations: continuous and approximation theories 1 Abstract parabolic systems
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| Format: | Book |
| Language: | English |
| Published: |
Cambridge
Cambridge Univ. Press
2010
|
| Edition: | 1. paperback ed. |
| Series: | Encyclopedia of mathematics and its applications
74 |
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| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXi, 644 S. |
| ISBN: | 9780521155670 |
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| 084 | |a SK 560 |0 (DE-625)143246: |2 rvk | ||
| 084 | |a MAT 356f |2 stub | ||
| 100 | 1 | |a Lasiecka, Irena |d 1948- |e Verfasser |0 (DE-588)111374383 |4 aut | |
| 245 | 1 | 0 | |a Control theory for partial differential equations |b continuous and approximation theories |n 1 |p Abstract parabolic systems |c Irena Lasiecka ; Roberto Triggiani |
| 250 | |a 1. paperback ed. | ||
| 264 | 1 | |a Cambridge |b Cambridge Univ. Press |c 2010 | |
| 300 | |a XXi, 644 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Encyclopedia of mathematics and its applications |v 74 | |
| 490 | 0 | |a Encyclopedia of mathematics and its applications |v ... | |
| 650 | 0 | 7 | |a Parabolische Differentialgleichung |0 (DE-588)4173245-5 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Parabolische Differentialgleichung |0 (DE-588)4173245-5 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Triggiani, Roberto |d 1942- |e Verfasser |0 (DE-588)112914292 |4 aut | |
| 773 | 0 | 8 | |w (DE-604)BV013028395 |g 1 |
| 830 | 0 | |a Encyclopedia of mathematics and its applications |v 74 |w (DE-604)BV000903719 |9 74 | |
| 856 | 4 | 2 | |m Digitalisierung UB Passau - ADAM Catalogue Enrichment |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027728647&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
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| adam_text | Contents
Preface
page
xv
0
Background
і
0.1
Some Function Spaces Used in Chapter
1 3
0.2
Regularity of the
Variation
of Parameter Formula When eAt
(s a
s.c. Analytic Semigroup
3
0.3
The Extrapolation Space
[7J(A::)ľ ó
0.4
Abstract Setting for Volume
L
The Operator
Lţ
m
(1.
1
,9),
or LsT
in
(1.4.1.6),
of Chapter
1 7
References and Bibliography
9
1
Optimal Quadratic Cost Problem Over a Preassigned Finite Time
Interval: Differential Riccati Equation
11
1.1
Mathematical Setting and Formulation of the Problem
12
1.2
Statement of Main Results
14
1.3
Orientation
21
1.4
Proof of Theorem
1.2.1.1
with GLT Closed
23
1.5
First Smoothing Case of the Operator
G
:
TheCase(-A*)^G*G
є
C(Y), ß >
2γ
- 1.
Proof of Theorem
1
.2.2.1 75
1.6
A Second Smoothing Case of the Operator G: The Case
(.....
А*У
G^G
€
£(F). Proof of Theorem
1.2.2.2 97
1.7
The Theory of Theorem
1.2.1.1
is Sharp. Counterexamples When
GLr
ís
Not Closabie
99
1.8
Extension to Unbounded Operators
R
and
G
103
1
A Proof of Lemma
1.5.1.1
(iii)
11
2
Notes on
Chápte»:
І
13
Ъ
Glossary of Symbols for Chapter
і
18
References and Bibliography
1
L9
2
Optimal Quadratic Cost Problem ovre.
яа ЛілНшїс Тїш« ЬмШ дпі
Algebraic Riccati Equatiou
ť/Á
2.1
Mathematical Setting and Formulation of the Problem
і
22
vu
viii Contents
2.2 Statement
of Main Results
125
2.3
Proof of
Theorem 2.2.1 129
2.4
Proof of Theorem
2.2.2:
Exponential Stability of
Ф(ѓ)
and
Uniqueness of the Solution of the Algebraic Riccati Equation
under the Detectability Condition
(2.1.13) 155
2.5
Extensions to Unbounded
R
:
R
є
£(Τ>(Αδ) Ζ),
5 <
min{l
-
γΑ]
160
2Α
Bounded Inversion of
[/ +
SV], S, V
> 0 167
2В
The Case
θ
= 1
in
(2.3.7.4)
When A is Self-Adjoint and
R
- / 168
Notes on Chapter
2 170
Glossary of Symbols for Chapter
2 175
References and Bibliography
176
3
Illustrations of the Abstract Theory of Chapters
1
and
2
ίο
Partial
Differential Equations with Boundary/Point Controls
178
3.0
Examples of Partial Differential Equation Problems Satisfying
Chapters
1
and
2 179
3.1
Heat Equation with Dirichlet Boundary Control: Riccati
Theory
180
3.2
Heat Equation with Dirichlet Boundary Control: Regularity
Theory of the Optimal Pair
187
3.3
Heat Equation with Neumann Boundary Control
194
3.4
A Structurally Damped Platelike Equation with Point Control and
Simplified Hinged
ВС
204
3.5
Kel
viri-Voight
Platelike Equation with Point Control with
Free
ВС
208
3.6
A Structurally Damped Platelike Equation with Boundary Control
in the Simplified Moment
ВС
2
1
í
3.7
Another Platelike Equation with Point Control and Clamped
ВС
214
3.8
The Strongly Damped Wave Equation with Point Control and
Dirichlet
ВС
216
3.9
A Structurally Damped
Kirchhoff
Equation with Point Control
Acting through
<$(·
-x0) and Simplified Hinged
ВС
218
ЗЛО
A Structurally Damped
Kirchhoff
Equation (Revisited) with Point
Control Acting through S (· -x°) and Simplified Hinged
ВС
221
3.11
Thermo-Elastic Plates with Thermal Control and Homogeneous
Clamped Mechanical
ВС
224
3.12
Thermo-Elastic Plates with Mechanical Control in the Bending
Moment (Hinged
ВС)
and Homogeneous Neumann Thermal
ВС
237
3.13
Thermo-Elastic Plates with Mechanical Control as a Shear Force
(Free
ВС)
248
3.14
Structurally Damped
Euler
-Bernoulli Equations with Damped
Free
ВС
and Point Control or Boundary Control
261
Contents ix
3.15
A Linearized Model of Well/Reservoir Coupling for a
Monophasic Flow with Boundary Control
269
3.16
Additional Illustrations with Control Operator
В
and Observation
Operator
R
Both Genuinely Unbounded
278
ЗА
Interpolation (Intermediate) Sobolev Spaces and Their
Identification with Domains of Fractional Powers of Elliptic
Operators
282
3B Damped Elastic Operators
285
3C Boundary Operators for Bending Moments and Shear Forces on
Two-Dimensional Domains
296
3D
Co-Semigroup/Analytic Semigroup Generation when A
—
AM, A
Positive Self-Adjoint,
M
Matrix. Applications to Thermo-Elastic
Equations with Hinged Mechanical
ВС
and Dirichlet Thermal
ВС
311
3E Analyticity of the s.c. Semigroups Arising from Abstract
Thermo- Elastic Equations. First Proof
324
3F Analyticity of the s.c. Semigroup Arising from Abstract
Thermo-Elastic Equations. Second Proof
346
3G Analyticity of the s.c. Semigroup Arising
nom
Abstract
Thermo-Elastic Equations. Third Proof
363
3H Analyticity of the s.c. Semigroup Arising from Problem
(3.12/1)
(Hinged Mechanical BC/Neumann (Robin) Thermal
ВС)
370
ЗІ
Analyticity of the s.c. Semigroup Arising from Problem
(3.13.1)
of
Section
13
(Free Mechanical BC/Neumann (Robin) Thermal
ВС)
382
3J Uniform Exponential Energy Decay of Thermo-Elastic Equations
with, or without, Rotational Term. Energy Methods
402
Notes on Chapter
3 413
References and Bibliography
425
Numerical Approximations of Algebraic Riccati Equations
431
4.1
Introduction: Continuous and Discrete Optimal Control Problems
431
4.2
Background Material
444
4.3
Convergence Properties of the Operators
L¡,
and
L*h; Ĺ/,
and L*h
446
4.4
Perturbation Results
451
4.5
Uniform Convergence Ph
í
ίΛ
-->■
P má
B¡¡PhUh
-■->·
B*P
471
4.6
Optimal Rates of Convergence
484
4
A A Sharp Result on the Exponential Operator-
Moi m
Decay of a
Family of Strongly Continuous Semigroups
488
4B Finite Element
Approximations
of Dynamic Compensators of
Luenberger s Type for Partially Observed Analytic Systems with
Fully Unbounded
Control
and Observation Operators 49 j
Notes on Chapter
4 504
Glossary of Symbols for Chapter
4
j09
References and Bibliography
509
χ
Contents
5
Illustrations
of the Numerical Theory of Chapter
4
to
Parabolic-Like Boundary/Point Control PDE Problems
511
5.1
Introductory Approximation Results
511
5.2
Heat Equation with Dirichlet Boundary Control
521
5.3
Heat Equation with Neumann Boundary Control. Optimal Rates
of Convergence with
r
> 1
and Galerkin Approximation
531
5.4
A Structurally Damped Platelike Equation with Interior Point
Control with
r
> 3 537
5.5
Kel
vin-Voight Platelike Equation with Interior Point Control with
r
> 3 544
5.6
A Structurally Damped Platelike Equation with Boundary Control
with
r
> 3 549
Notes on Chapter
5 554
Glossary of Symbols for Chapter
5,
Section
5.1 554
References and Bibliography
554
6
Min-Max Game
Theory over an Infinite Time interval and
Algebraic Riccati Equations
556
Part I: General Case
557
6.1
Mathematical Setting; Formulation of the
Min-Max Game
Problem; Statement of Main Results
557
6.2
Minimization of JWtr over
и
є
/^(O,
T; U)
for
w
Fixed
562
6.3
Minimization of
ЈШлОС
over
и
є
/,2(0,
σο;
U)
for
w
Fixed: The
Limit Process
asľ f
oů 570
6.4
Collection of Explicit Formulae for pw,oo,
Voo
and y£jOO in
Stable Form
581
6.5
Explicit Expression for the Optimal Cost j£
^(уо
= 0)
as a
Quadratic Term
583
6.6
Definition of the Critical Value yc, Coercivity of Ey for
у
>
yc
585
6.7
Maximization of J® ^ over
ш
Directly on
[0,
od]
for
γ
>
yc.
Characterization of Optimal Quantities
586
6.8
Explicit Expression of w*
(· ;
yo)
in Terms of the Data via Ey~l for
У > Ус
589
6.9
Smoothing Properties of the Operators
Ĺ,
/,*,
W, W*: The
Optimal
m*,
y
ш*
Are Continuous in Time
589
6.10
A Transition Property for w* for
γ
>
yc
593
6.11
A Transition Property for
r
for
γ
>
yc
S9ÍJ
6.12
The Semigroup Property for y* and a Transition Property for /r:
for
у
>
yc
596
6.13
Definition of
Ρ
and Its Properties
598
6.14
The Feedback Generator A p and Its Preliminary Properties for
γ
>
rc
600
Contents xi
6.15 The Operator
Ρ
is a Solution of the Algebraic Riccati Equation,
AREy for
y
>
yc
603
6.16
The Semigroup Generated by (A ~ BB* P) Is Uniformly Stable
604
6.17
The Case
0 <
у
<
yc sup
j£ì00(yo) =
+00 606
6.18
Proof of Theorem
6.1.3.2 607
Part II: The Case Where eAt is Stable
608
6.19
Motivation, Statement of Main Results
608
6.20
Minimization of
J
over
и
for
w
Fixed
612
6.21
Maximization of J®(yo) over w Existence of a Unique
Optimal w*
616
6.22
Explicit Expressions of
{«*,
y*, w*} and
P
for
у
>
yc in Terms
of the Data via E~l
618
6.23
Smoothing Properties of the Operators
ƒ. ,,
¿Д
W, W *: The
Optimal
ir
,
y% w* Are Continuous in Time
620
6.24
A Transition Property for
w
i:; for
у
>-
^c
6.25
The Semigroup Property for
у
:: for
у
>
кс
and its Stability
6.26
The Riccati Operator, P, for
y > yĽ
627
6A Optimal Control Problem with Nondefimte Quadratic Cost. The
Stable, Analytic Case. A Brief Sketch
630
Notes on Chapter
6 639
References and Bibliography
642
Contents of Volume II
7
Some Auxiliary Results on Abstract Equations
645
7.1
Mathematical Setting and Standing Assumptions
645
7.2
Regularity of
L
and L* on
[0,
T]
648
7.3
A Lifting Regularity Property When eAt Is a Group
651
7.4
Extension of Regularity of
L
and L* on
[0, 00]
When eAt
ís
Uniformly Stable
653
7.5
Generation and Abstract Trace Regularity under Unbounded
Perturbation
660
7.6
Regularity of a Class of Abstract Damped Systems
663
7.7
Illustrations of Theorem
7,6.2.2
to
Boundary Damped Wave
Hquations
667
Notes on Chapter
7 671
References and Bibliography
67!
З Оріішш
Quaâmíic
Cosí: líVofoSem Cvsr a
irs^sssśgiiseii
-¿ Зші?, лИнійс
Interval: The Case Wher«
Èïin
ϊιιμίϊί:
-і-
SoIkoorí. iVm)~
їз
Unbounded, but the input
■ ->■
Observation Map Is
öoimdoí .
-ľ/ ]
8.1
Mathematical Setting and Foimulaiioii of the
РгоЫеш
6/5
8.2
Statement of Main Results
6/9
xii Contents
8.3 The General
Case. A First Proof of Theorems
8.2.1.1
and
8.2.1.2
by a Variational Approach: From the Optimal Control Problem
to the
DRE
and the IRE Theorem
8.2.1.3 687
8.4
A Second Direct Proof of Theorem
8.2.1.2:
From the
Well-Posedness of the IRE to the Control Problem. Dynamic
Programming
714
8.5
Proof of Theorem
8.2.2.1 :
The More Regular Case
733
8.6
Application of Theorems
8.2.1.1, 8.2.
í
.2,
and
8.2.2.1 :
Neumann
Boundary Control and Dirichlet Boundary Observation for
Second-Order Hyperbolic Equations
736
8.7
A One-Dimensional Hyperbolic Equation with Dirichlet Control
(B Unbounded) and Point Observation (R Unbounded) That
Satisfies (h.
1)
and (h.3) but not (h.2), (H.
1 ),
(H.2), and (H.3). Yet,
the
DRE
Is Trivially Satisfied as a Linear Equation
745
8
A interior and Boundary Regularity of Mixed Problems for
Second-Order Hyperbolic Equations with Neumann-Type
ВС
755
Notes on Chapter
8 /61
References and Bibliography
/63
9
Optimal Quadratic Cost Problem over a Preassigiied Finite Time
Interval: The Case Where the Input
--■>
Solution Map Is Bounded»
Differential and
Integrai
Riccati
Equations
765
9.1
Mathematical Setting and Formulation of the Problem
765
9.2
Statement of Main Result: Theorems
9.2.1, 9.2.2,
and
9.2.3 772
9.3
Proofs of Theorem
9.2.1
and Theorem
9.2.2
(by the Variational
Approach and by the Direct Approach). Proof of Theorem
9.2.3 776
9.4
Isomorphism of P(t),
0 <
t
<
T,
and Exact Controllability of
{A*, R*} on
[0,
T
- /]
When
G
== 0 815
9.5
Nonsmoothing Observation R: Limit Solution of the
Differential Riccati Equation under the Sole Assumption (A.I)
When
G
-=0 819
9.6
Dual Differential and Intergral Riccati Equations When
Л
is a
Group Generator under (A.I) and
R
є
C(Y;
Z)
and
G
~
0.
(Bounded Control Operator, Unbounded Observation)
825
9.7
Optimal Control Problem with Bounded Control Operator and
Unbounded Observation Operator
839
9.8
Application to Hyperbolic Partial Differential Equations with Point
Control. Regularity Theory
842
9.9
Proof of Regularity Results Needed in Section
9.8 861
9.10
A Coupled System of a Wave and
a
Kirchhoff
Equation with Point
Control, Arising in Noise Reduction. Regularity Theory
884
Contents xiii
9.1 ]
A Coupled System of a Wave and a Structurally Damped
Euler-Bernoulli Equation with Point Control, Arising in Noise
Reduction, Regularity Theory
901
9A Proof of
(9.9.1.16)
in Lemma
9.9.1.1 908
9B Proof of
(9.9.3.14)
in Lemma
9.9.3.1 910
Notes on Chapter
9 913
References and Bibliography
916
10
Differential Riccati Equations under Slightly Smoothing
Observation Operator. Applications to Hyperbolic and
Petro
wski-Type PDEs. Regularity Theory
9Ï9
10.1
Mathematical Setting and Problem Statement
920
10.2
Statement of the Main Results
926
10.3
Proof of Theorems
10.2.1
and
10.2.2 928
10.4
Proof of Theorem
10.2.3 936
10.5
Application: Second-Order Hyperbolic Equations with Dirichlet
Boundary Control. Regularity Theory
942
10.6
Application: Nonsymmetric, Nondissipative
First-Oťdeí
Hyperbolic Systems with Boundary Control. Regularity Theory
972
10.7
Application:
Kirchoff
Equation with One Boundary Control.
Regularity Theory
989
10.8
Application: Euler-Bernoulli Equation with One Boundary
Control. Regularity Theory
1019
10.9
Application:
Schrödinger
Equations with Dirichlet Boundary
Control. Regularity Theory
1042
Notes on Chapter
10 1059
Glossary of Selected Symbols for Chapter
10 1065
References and Bibliography
1065
Index
|
| any_adam_object | 1 |
| author | Lasiecka, Irena 1948- Triggiani, Roberto 1942- |
| author_GND | (DE-588)111374383 (DE-588)112914292 |
| author_facet | Lasiecka, Irena 1948- Triggiani, Roberto 1942- |
| author_role | aut aut |
| author_sort | Lasiecka, Irena 1948- |
| author_variant | i l il r t rt |
| building | Verbundindex |
| bvnumber | BV042291483 |
| classification_rvk | SK 560 |
| classification_tum | MAT 356f |
| ctrlnum | (OCoLC)907835838 (DE-599)BVBBV042291483 |
| discipline | Mathematik |
| edition | 1. paperback ed. |
| format | Book |
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| id | DE-604.BV042291483 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T01:17:27Z |
| institution | BVB |
| isbn | 9780521155670 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-027728647 |
| oclc_num | 907835838 |
| open_access_boolean | |
| owner | DE-739 |
| owner_facet | DE-739 |
| physical | XXi, 644 S. |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| series | Encyclopedia of mathematics and its applications |
| series2 | Encyclopedia of mathematics and its applications |
| spelling | Lasiecka, Irena 1948- Verfasser (DE-588)111374383 aut Control theory for partial differential equations continuous and approximation theories 1 Abstract parabolic systems Irena Lasiecka ; Roberto Triggiani 1. paperback ed. Cambridge Cambridge Univ. Press 2010 XXi, 644 S. txt rdacontent n rdamedia nc rdacarrier Encyclopedia of mathematics and its applications 74 Encyclopedia of mathematics and its applications ... Parabolische Differentialgleichung (DE-588)4173245-5 gnd rswk-swf Parabolische Differentialgleichung (DE-588)4173245-5 s DE-604 Triggiani, Roberto 1942- Verfasser (DE-588)112914292 aut (DE-604)BV013028395 1 Encyclopedia of mathematics and its applications 74 (DE-604)BV000903719 74 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027728647&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lasiecka, Irena 1948- Triggiani, Roberto 1942- Control theory for partial differential equations continuous and approximation theories Encyclopedia of mathematics and its applications Parabolische Differentialgleichung (DE-588)4173245-5 gnd |
| subject_GND | (DE-588)4173245-5 |
| title | Control theory for partial differential equations continuous and approximation theories |
| title_auth | Control theory for partial differential equations continuous and approximation theories |
| title_exact_search | Control theory for partial differential equations continuous and approximation theories |
| title_full | Control theory for partial differential equations continuous and approximation theories 1 Abstract parabolic systems Irena Lasiecka ; Roberto Triggiani |
| title_fullStr | Control theory for partial differential equations continuous and approximation theories 1 Abstract parabolic systems Irena Lasiecka ; Roberto Triggiani |
| title_full_unstemmed | Control theory for partial differential equations continuous and approximation theories 1 Abstract parabolic systems Irena Lasiecka ; Roberto Triggiani |
| title_short | Control theory for partial differential equations |
| title_sort | control theory for partial differential equations continuous and approximation theories abstract parabolic systems |
| title_sub | continuous and approximation theories |
| topic | Parabolische Differentialgleichung (DE-588)4173245-5 gnd |
| topic_facet | Parabolische Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027728647&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV013028395 (DE-604)BV000903719 |
| work_keys_str_mv | AT lasieckairena controltheoryforpartialdifferentialequationscontinuousandapproximationtheories1 AT triggianiroberto controltheoryforpartialdifferentialequationscontinuousandapproximationtheories1 |