Control theory for partial differential equations: continuous and approximation theories 2 Abstract hyperbolic-like systems over a finite time horizon
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge
Cambridge Univ. Press
2010
|
| Ausgabe: | 1. paperback ed. |
| Schriftenreihe: | Encyclopedia of mathematics and its applications
75 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XXI, S. 645 - 1067 |
| ISBN: | 9780521155687 |
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MARC
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|---|---|---|---|
| 001 | BV042291497 | ||
| 003 | DE-604 | ||
| 005 | 20150407 | ||
| 007 | t | ||
| 008 | 150126s2010 |||| 00||| eng d | ||
| 020 | |a 9780521155687 |9 9780521155687 | ||
| 035 | |a (OCoLC)907780250 | ||
| 035 | |a (DE-599)BVBBV042291497 | ||
| 040 | |a DE-604 |b ger |e rakwb | ||
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| 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 2 |p Abstract hyperbolic-like systems over a finite time horizon |c Irena Lasiecka ; Roberto Triggiani |
| 250 | |a 1. paperback ed. | ||
| 264 | 1 | |a Cambridge |b Cambridge Univ. Press |c 2010 | |
| 300 | |a XXI, S. 645 - 1067 | ||
| 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 75 | |
| 490 | 0 | |a Encyclopedia of mathematics and its applications |v ... | |
| 650 | 0 | 7 | |a Hyperbolische Differentialgleichung |0 (DE-588)4131213-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Parabolische Differentialgleichung |0 (DE-588)4173245-5 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Hyperbolische Differentialgleichung |0 (DE-588)4131213-2 |D s |
| 689 | 0 | |5 DE-604 | |
| 689 | 1 | 0 | |a Parabolische Differentialgleichung |0 (DE-588)4173245-5 |D s |
| 689 | 1 | |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 2 |
| 830 | 0 | |a Encyclopedia of mathematics and its applications |v 75 |w (DE-604)BV000903719 |9 75 | |
| 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=027728660&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-027728660 | ||
Datensatz im Suchindex
| _version_ | 1804152862530338816 |
|---|---|
| adam_text | Contents
Preface
page
xv
7
Some Auxiliary Results on Abstract Equations 54S
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
ел
is a Group
651
7.4
Extension of Regularity of
L
and //:: on
[0,
сю]
When eAt is
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
Equations
667
Notes on Chapter
7 671
References and Bibliography
671
8
Optimal Quadratic Cost Problem Over a Preassigned Finite Time
Interval: The Case Where the Input
->
Solution Map Is
Unbounded, but the Input
—*■
Observation Map Is Bounded
673
8.1
Mathematical Setting and Formulation of the Problem
675
8.2
Statement of Main Results
679
8.3
The General Case. A First Proof of Theorems
8.2, !,1
and
8.2.1.2
by a Variational Approach: From the Optimal Control Problem
to the
DRE
and the IRE Theorem
8.2. ) .3 687
8.4
A Second Direct Proof of Theorem
8.2.1,2:
From the
Well-Posedness of the IRE
řo
the Control Problem. Dynamic
Programming
714
j
8.5
Proof of Theorem
8.2.2.1 :
The More Regular
(
Jase
733
8.6
Application of Theorems
8.2.1.1, 8.2.1.2,
and
8.2.2.1:
Neumami
Boundary Control and Dirichlet Boundary Observation for
Second-Order Hyperbolic Equations
736
VII
viii Contents
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.I), (H.2), and (H.3). Yet,
the
DRE
Is Trivially Satisfied as a Linear Equation
745
8A Interior and Boundary Regularity of Mixed Problems for
Second-Order Hyperbolic Equations with Neumann-Type
ВС
755
Notes on Chapter
8 761
References and Bibliography
763
9
Optimal Quadratic Cost Problem over a Preassigned Finite Time
Interval: The Case Where the Input
->
Solution Map Is Bounded.
Differential and Integral 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*,
/?*}
on
[0,
T
-
t]
When
G
^
0 815
9.5
Nonsmoothing Observation R: Limit Solution of the
Differential Riccati Equation under the Sole Assumption (A.
1 )
When
G
=0 819
9.6
Dual Differential and Intergral Riccati Equations When A 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
9.11
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 Operatoi; Applications to Hyperbolic and
Petrowski-Type PDE. ,. Regularity Theory
919
10.1
Mathematical Setting and Problem Statement
920
Contents ix
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-Order
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 ;i
065
Contents
οι
Volume
1
0
Background
il
0.1
Some Function Spaces Used
iu
Chapter
1 3
0.2
Regularity of the Variation of Parameter Formula When eAt Is a
s.c. Analytic Semigroup
3
0.3
The Extrapolation Space [ViA*)]
6
0.4
Abstract Setting for Volume I. The Operator
L
T in
(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
G L·
т
Closed
73
1.5
First Smoothing Case of the Operator G: The Case
(■
·Α *γθ*Ό
с.
£(Г),
β
>
2γ
■ ■
і.
Proof or Theorem
1.2.7..
і
I S
1.6
A Second Smoothing Case of the Operator G: The Case
(-■■A*)* G*G
с-
C(Y). Proof of Theorem
Ι. λ. Λ. Λ
97
1.7
The Theory of
«
Ъеогет
1. ?..
. 1
Is Sharp. Counterexamples When
G L r
Is Not Closable
99
1.8
Extension to Unbounded Operators
R
and
G
103
1
A Proof of Lemma
1.5.1.
Ціп)
112
Notes on Chapter
1 113
x
Contents
Glossary of Symbols for Chapter
1 118
References and Bibliography
119
2
Optimal Quadratic Cost Problem over an Infinite Time Interval:
Algebraic Riccati Equation
121
2.1
Mathematical Setting and Formulation of the Problem
122
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
Φ
(t)
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 СфіА5)]
Z),
S
<
min{l
-γΑ}
160
2A
Bounded Inversion of
[I -f· SV], S, V
> 0 167
2B The Case
0
^
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
to 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
1
87
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
Kelvin-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
211
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 <5(
.....
л 0)
and Simplified Hinged
ВС
218
3.10
A Structurally Damped
Kirchhoff
Equation (Revisited) with Point
Control Acting through
б (
.....
x°) and Simplified Hinged
ВС
221
3.11
Thermo-Elastic Plates with Thermal Control and Homogeneous
Clamped Mechanical
ВС
224
Contents xi
З
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 EuJer-Bernoulli Equations with Damped
Free
ВС
and Point Control or Boundary Control
261
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 ou
Two-Dìmeiisional
Domains 296
3D
Co-Semi group/Analy tic Semigroup Generation when A
■ ■■
AM, A
Positive Self-Adjoint,
M
Matrix. Applications to Theraio-Elastic
Equations with Hinged Mechanical
ВС
and Dirichlel 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 from 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
ВС
/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 4
Î
3
References and Bibliography
423
Numerical Approximations
©ίΓ
Algebróte
Миссий Есртйкюе
dV«
4.1
Introduction: Continuous and Discrete Optimal Control Problems
43
і
4.2
Background Material
Ą4A
4.3
Convergence Properties of the Operators Lh and l,% Lh and
ÎA
446
4.4
Perturbation Results
451
4.5
Uniform Convergence
РдП/,
··>
Ρ
and
BţPh h
>
Β* Ρ
471
4.6
Optimal
Rates of Convergence
484
4
A A Sharp Result on the Exponential Operator Norm Decay of a
Family of Strongly Continuous Semigroups
488
xii Contents
4В
Finite
Element
Approximations
of Dynamic Compensators of
Luenberger s Type for Partially Observed Analytic Systems with
Fully Unbounded Control and Observation Operators
495
Notes on Chapter
4 504
Glossary of Symbols for Chapter
4 509
References and Bibliography
509
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
г
> 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
г
> 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
Jw,ţ
over
и
є
¿2(0,
T; U)
for w Fixed
562
6.3
Minimization of
Јш,сс
over
и
є
2-2(0,
oo;
U)
for
w
Fixed: The
Eimit Process as
T t
oo
570
6.4
Collection of Explicit Formulae for pWiOO, rw%oc, and y%i0O in
Stable Form
581
6.5
Explicit Expression for the Optimal Cost J® ;X5( > o
0)
as a
Quadratic Term
583
6.6
Definition of the Critical Value yc. Coercivity of EY for
γ
>
yc
585
6.7
Maximization of J®iOO over
w
Directly oo
[0,
oo] for
γ
>
yc,
Characterization of Optimal Quantities
586
6.
о
Explicit Expression of w*(·
;
yo)
m
Terms of the Data via Eyl for
γ
>
Yc
589
6.9
Smoothing Properties of the Operators
Ĺ, Ĺ*,
W, W*:
The
Optimal
іґ
,
у*,
w*
Are Continuous in Time
589
Contents xiii
6.10
A Transition Property for w* for
γ
>
yc
593
6.11
A Transition Property for r* for
γ
>
yc
595
6.12
The Semigroup Property for y* and a Transition Property for p*
for
γ
>
Yc
596
6.13
Definition of
Ρ
and Its Properties
598
6.14
The Feedback Generator AF and Its Preliminary Properties for
γ
>
Yc
600
6.15
The Operator
Ρ
is a Solution of the Algebraic Riccati Equation,
AREy for
γ
>
Yc
603
6.16
The Semigroup Generated by (A
—
BB* Iу) Is Uniformly Stable
604
6.17
The Case
0 <
γ
<
yc sup J£t00(yo)
— fco 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®(} o)
ovei1
ш:
Existence of a Unique
Optimal
Ш*
uio
6.22
Explicit Expressions of
{«*,
y;:, ur} and /:> for
у
>
yc in Terms
of the Data via E 1
618
6.23
Smoothing Properties of the Operators
Ĺ,
V
,
IV, W*: The
Optimal
и*,
)>*,
ш*
Are Continuous in Time
620
6.24
A Transition Property for w* for
у
>
j/c
622
6.25
The Semigroup Property for y* for
у
>
yc and Its Stability
626
6.26
The Riccati Operator, P, for
γ
>
)/c
627
6A Optimal Control Problem with
Nondefinite
Quadratic Cost. The
Stable, Analytic Case. A Brief Sketch
630
Notes on Chapter
6 639
References and Bibliography
642
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 | BV042291497 |
| classification_rvk | SK 560 |
| classification_tum | MAT 356f |
| ctrlnum | (OCoLC)907780250 (DE-599)BVBBV042291497 |
| discipline | Mathematik |
| edition | 1. paperback ed. |
| format | Book |
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| id | DE-604.BV042291497 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T01:17:27Z |
| institution | BVB |
| isbn | 9780521155687 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-027728660 |
| oclc_num | 907780250 |
| open_access_boolean | |
| owner | DE-739 DE-20 |
| owner_facet | DE-739 DE-20 |
| physical | XXI, S. 645 - 1067 |
| 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 2 Abstract hyperbolic-like systems over a finite time horizon Irena Lasiecka ; Roberto Triggiani 1. paperback ed. Cambridge Cambridge Univ. Press 2010 XXI, S. 645 - 1067 txt rdacontent n rdamedia nc rdacarrier Encyclopedia of mathematics and its applications 75 Encyclopedia of mathematics and its applications ... Hyperbolische Differentialgleichung (DE-588)4131213-2 gnd rswk-swf Parabolische Differentialgleichung (DE-588)4173245-5 gnd rswk-swf Hyperbolische Differentialgleichung (DE-588)4131213-2 s DE-604 Parabolische Differentialgleichung (DE-588)4173245-5 s Triggiani, Roberto 1942- Verfasser (DE-588)112914292 aut (DE-604)BV013028395 2 Encyclopedia of mathematics and its applications 75 (DE-604)BV000903719 75 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=027728660&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 Hyperbolische Differentialgleichung (DE-588)4131213-2 gnd Parabolische Differentialgleichung (DE-588)4173245-5 gnd |
| subject_GND | (DE-588)4131213-2 (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 2 Abstract hyperbolic-like systems over a finite time horizon Irena Lasiecka ; Roberto Triggiani |
| title_fullStr | Control theory for partial differential equations continuous and approximation theories 2 Abstract hyperbolic-like systems over a finite time horizon Irena Lasiecka ; Roberto Triggiani |
| title_full_unstemmed | Control theory for partial differential equations continuous and approximation theories 2 Abstract hyperbolic-like systems over a finite time horizon 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 hyperbolic like systems over a finite time horizon |
| title_sub | continuous and approximation theories |
| topic | Hyperbolische Differentialgleichung (DE-588)4131213-2 gnd Parabolische Differentialgleichung (DE-588)4173245-5 gnd |
| topic_facet | Hyperbolische Differentialgleichung Parabolische Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027728660&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 controltheoryforpartialdifferentialequationscontinuousandapproximationtheories2 AT triggianiroberto controltheoryforpartialdifferentialequationscontinuousandapproximationtheories2 |