Control Theory for Partial Differential Equations: Continuous and Approximation Theories

by Irena Lasiecka , Roberto Triggiani

• ISBN13: 9780521584012
• ISBN10: 0521584019
• Format: Hardcover
• Copyright: 2000-02-13
• Publisher: Cambridge University Press

Summary

This is the second volume of a comprehensive and up-to-date two-volume treatment of quadratic optimal control theory for partial differential equations over a finite or infinite time horizon, and related differential (integral) and algebraic Riccati equations. Both continuous theory and numerical approximation theory are included. The authors use an abstract space, operator theoretic approach, which is based on semigroups methods, and which unifies across a few basic classes of evolution. The various abstract frameworks are motivated by, and ultimately directed to, partial differential equations with boundary/point control. Volume 2 is focused on the optimal control problem over a finite time interval for hyperbolic dynamical systems. A few abstract models are considered, each motivated by a particular canonical hyperbolic dynamics. It presents numerous new results. These volumes will appeal to graduate students and researchers in pure and applied mathematics and theoretical engineering with an interest in optimal control problems.

Table of Contents

Preface	page xv
Some Auxiliary Results on Abstract Equations	645	(28)
Mathematical Setting and Standing Assumptions	645	(3)
Regularity of L and L* on [0,T]	648	(3)
A Lifting Regularity Property When eAt Is a Group	651	(2)
Extension of Regularity of L and L* on [0, ∞] When eAt Is Uniformly Stable	653	(7)
Generation and Abstract Trace Regularity under Unbounded Perturbation	660	(3)
Regularity of a Class of Abstract Damped Systems	663	(4)
Illustrations of Theorem 7.6.2.2 to Boundary Damped Wave Equations	667	(6)
Notes on Chapter 7	671	(1)
References and Bibliography	671	(2)
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	(92)
Mathematical Setting and Formulation of the Problem	675	(4)
Statement of Main Results	679	(8)
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	(27)
A Second Direct Proof of Theorem 8.2.1.2: From the Well-Posedness of the IRE to the Control Problem. Dynamic Programming	714	(19)
Proof of Theorem 8.2.2.1: The More Regular Case	733	(3)
Application of Theorems 8.2.1.1, 8.2.1.2, and 8.2.2.1: Neumann Boundary Control and Dirichlet Boundary Observation for Second-Order Hyperbolic Equations	736	(9)
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	(10)
Interior and Boundary Regularity of Mixed Problems for Second-Order Hyperbolic Equations with Neumann-Type BC	755	(10)
Notes on Chapter 8	761	(2)
References and Bibliography	763	(2)
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	(154)
Mathematical Setting and Formulation of the Problem	765	(7)
Statement of Main Result: Theorems 9.2.1, 9.2.2, and 9.2.3	772	(4)
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	(39)
Isomorphism of P(t), 0 ≤ t < T, and Exact Controllability of [A*, R*] on [0, T -- t] When G = 0	815	(4)
Nonsmoothing Observation R: ``Limit Solution'' of the Differential Riccati Equation under the Sole Assumption (A.1) When G = 0	819	(6)
Dual Differential and Intergral Riccati Equations When A is a Group Generator under (A.1) and R ∈ ℒ (Y; Z) and G = 0. (Bounded Control Operator, Unbounded Observation)	825	(14)
Optimal Control Problem with Bounded Control Operator and Unbounded Observation Operator	839	(3)
Application to Hyperbolic Partial Differential Equations with Point Control. Regularity Theory	842	(19)
Proof of Regularity Results Needed in Section 9.8	861	(23)
A Coupled System of a Wave and a Kirchhoff Equation with Point Control, Arising in Noise Reduction. Regularity Theory	884	(17)
A Coupled System of a Wave and a Structurally Damped Euler--Bernoulli Equation with Point Control, Arising in Noise Reduction. Regularity Theory	901	(7)
Proof of (9.9.1.16) in Lemma 9.9.1.1	908	(2)
Proof of (9.9.3.14) in Lemma 9.9.3.1	910	(9)
Notes on Chapter 9	913	(3)
References and Bibliography	916	(3)
Differential Riccati Equations under Slightly Smoothing Observation Operator. Applications to Hyperbolic and Petrowski-Type PDEs. Regularity Theory	919
Mathematical Setting and Problem Statement	920	(6)
Statement of the Main Results	926	(2)
Proof of Theorems 10.2.1 and 10.2.2	928	(8)
Proof of Theorem 10.2.3	936	(6)
Application: Second-Order Hyperbolic Equations with Dirichlet Boundary Control. Regularity Theory	942	(30)
Application: Nonsymmetric, Nondissipative First-Order Hyperbolic Systems with Boundary Control. Regularity Theory	972	(17)
Application: Kirchoff Equation with One Boundary Control. Regularity Theory	989	(30)
Application: Euler-Bernoulli Equation with One Boundary Control. Regularity Theory	1019	(23)
Application: Schrodinger Equations with Dirichlet Boundary Control. Regularity Theory	1042
Notes on Chapter 10	1059	(6)
Glossary of Selected Symbols for Chapter 10	1065	(1)
References and Bibliography	1065
Contents of Volume I
Background	1	(10)
Some Function Spaces Used in Chapter 1	3	(1)
Regularity of the Variation of Parameter Formula When eAt Is a s.c. Analytic Semigroup	3	(3)
The Extrapolation Space [D(A*)]'	6	(1)
Abstract Setting for Volume I. The Operator LT in (1.1.9), or LsT in (1.4.1.6), of Chapter 1	7	(4)
References and Bibliography	9	(2)
Optimal Quadratic Cost Problem Over a Preassigned Finite Time Interval: Differential Riccati Equation	11	(110)
Mathematical Setting and Formulation of the Problem	12	(2)
Statement of Main Results	14	(7)
Orientation	21	(2)
Proof of Theorem 1.2.1.1 with GLT Closed	23	(52)
First Smoothing Case of the Operator G: The Case (-- A*)β G* G ∈ ℒ(Y), β > 2γ -- 1. Proof of Theorem 1.2.2.1	75	(22)
A Second Smoothing Case of the Operator G: The Case (-- A*)γ G*G ∈ ℒ(Y). Proof of Theorem 1.2.2.2	97	(2)
The Theory of Theorem 1.2.1.1 Is Sharp. Counterexamples When GLT Is Not Closable	99	(4)
Extension to Unbounded Operators R and G	103	(9)
Proof of Lemma 1.5.1.1(iii)	112	(9)
Notes on Chapter 1	113	(5)
Glossary of Symbols for Chapter 1	118	(1)
References and Bibliography	119	(2)
Optimal Quadratic Cost Problem over an Infinite Time Interval: Algebraic Riccati Equation	121	(57)
Mathematical Setting and Formulation of the Problem	122	(3)
Statement of Main Results	125	(4)
Proof of Theorem 2.2.1	129	(26)
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	(5)
Extensions to Unbounded R : R ∈ ℒ(D (Aδ); Z), δ < min {1 -- γ, 1/2}	160	(7)
Bounded Inversion of [I + SV], S, V ≥ 0	167	(1)
The Case &thetas; = 1 in (2.3.7.4) When A is Self-Adjoint and R = I	168	(10)
Notes on Chapter 2	170	(5)
Glossary of Symbols for Chapter 2	175	(1)
References and Bibliography	176	(2)
Illustrations of the Abstract Theory of Chapters 1 and 2 to Partial Differential Equations with Boundary/Point Controls	178	(253)
Examples of Partial Differential Equation Problems Satisfying Chapters 1 and 2	179	(1)
Heat Equation with Dirichlet Boundary Control: Riccati Theory	180	(7)
Heat Equation with Dirichlet Boundary Control: Regularity Theory of the Optimal Pair	187	(7)
Heat Equation with Neumann Boundary Control	194	(10)
A Structurally Damped Platelike Equation with Point Control and Simplified Hinged BC	204	(4)
Kelvin-Voight Platelike Equation with Point Control with Free BC	208	(3)
A Structurally Damped Platelike Equation with Boundary Control in the Simplified Moment BC	211	(3)
Another Platelike Equation with Point Control and Clamped BC	214	(2)
The Strongly Damped Wave Equation with Point Control and Dirichlet BC	216	(2)
A Structurally Damped Kirchhoff Equation with Point Control Acting through δ(. --x0) and Simplified Hinged BC	218	(3)
A Structurally Damped Kirchhoff Equation (Revisited) with Point Control Acting through δ'(. --x0) and Simplified Hinged BC	221	(3)
Thermo-Elastic Plates with Thermal Control and Homogeneous Clamped Mechanical BC	224	(13)
Thermo-Elastic Plates with Mechanical Control in the Bending Moment (Hinged BC) and Homogeneous Neumann Thermal BC	237	(11)
Thermo-Elastic Plates with Mechanical Control as a Shear Force (Free BC)	248	(13)
Structurally Damped Euler-Bernoulli Equations with Damped Free BC and Point Control or Boundary Control	261	(8)
A Linearized Model of Well/Reservoir Coupling for a Monophasic Flow with Boundary Control	269	(9)
Additional Illustrations with Control Operator B and Observation Operator R Both Genuinely Unbounded	278	(4)
Interpolation (Intermediate) Sobolev Spaces and Their Indentification with Domains of Fractional Powers of Elliptic Operators	282	(3)
Damped Elastic Operators	285	(11)
Boundary Operators for Bending Moments and Shear Forces on Two-Dimensional Domains	296	(15)
C0-Semigroup/Analytic Semigroup Generation when A = AM, A Positive Self-Adjoint, M Matrix. Applications to Thermo-Elastic Equations with Hinged Mechanical BC and Dirichlet Thermal BC	311	(13)
Analyticity of the s.c. Semigroups Arisng from Abstract Thermo-Elastic Equations. First Proof	324	(22)
Analyticity of the s.c. Semigroup Arising from Abstract Thermo-Elastic Equations. Second Proof	346	(17)
Analyticity of the s.c. Semigroup Arising from Abstract Thermo-Elastic Equations. Third Proof	363	(7)
Analyticity of the s.c. Semigroup Arising from Problem (3.12.1) (Hinged Mechanical BC/Neumann (Robin) Thermal BC)	370	(32)
Uniform Exponential Energy Decay of Thermo-Elastic Equations with, or without, Rotational Term. Energy Methods	402	(29)
Notes on Chapter 3	413	(12)
References and Bibliography	425	(6)
Numerical Approximations of Algebraic Riccati Equations	431	(80)
Introduction: Continuous and Discrete Optimal Control Problems	431	(1)
Background Material	431	(15)
Convergence Properties of the Operators Lh and Last;h; Lh and L*h	446	(5)
Perturbation Results	451	(20)
Uniform Convergence Ph Πh → P and B*h Ph Πh → B* P	471	(13)
Optimal Rates of Convergence	484	(4)
A Sharp Result on the Exponential Operator-Norm Decay of a Family of Strongly Continuous Semigroups	488	(7)
Finite Element Approximations of Dynamic Compensators of Luenberger's Type for Partially Observed Analytic Systems with Fully Unbounded Control and Observation Operators	495	(16)
Notes on Chapter 4	504	(5)
Glossary of Symbols for Chapter 4	509	(1)
References and Bibliography	509	(2)
Illustrations of the Numerical Theory of Chapter 4 to Parabolic-Like Boundary/Point Control PDE Problems	511	(45)
Introductory Approximation Results	511	(10)
Heat Equation with Dirichlet Boundary Control	521	(10)
Heat Equation with Neumann Boundary Control. Optimal Rates of Convergence with r ≥ 1 and Galerkin Approximation	531	(6)
A Structurally Damped Platelike Equation with Interior Point Control with r ≥ 3	537	(7)
Kelvin--Voight Platelike Equation with Interior Point Control with r ≥ 3	544	(5)
A Structurally Damped Platelike Equation with Boundary Control with r ≥ 3	549	(7)
Notes on Chapter 5	554	(1)
Glossary of Symbols for Chapter 5, Section 5.1	554	(1)
References and Bibliography	554	(2)
Min--Max Game Theory over an Infinite Time Interval and Algebraic Riccati Equations	556
Part I: General Case	557	(1)
Mathematical Setting; Formulation of the Min-Max Game Problem; Statement of Main Results	557	(5)
Minimization of Jw,T over u ∈ L2(0, T; U) for w Fixed	562	(8)
Minimization of Jw,∞ over u ∈ L2(0, ∞; U) for w Fixed: The Limit Process as T ↑ ∞	570	(11)
Collection of Explicit Formulae for pw,∞, rw,∞, and y0w,∞ in Stable Form	581	(2)
Explicit Expression for the Optimal Cost J0w,∞ (y0 = 0) as a Quadratic Term	583	(2)
Definition of the Critical value γc. Coercivity of Eγ for γ > γc	585	(1)
Maximization of J0w,∞ over w Directly on [0, ∞] for γ > γc. Characterization of Optimal Quantities	586	(3)
Explicit Expression of w*(.; y0) in Terms of the Data via E--1γ for γ > γc	589	(1)
Smoothing Properties of the Operators L, L*, W, W*: The Optimal u*, y*, w* Are Continuous in Time	589	(4)
A Transition Property for w* for γ > γc	593	(2)
A Transition Property for r* for γ > γc	595	(1)
The Semigroup Property for y* and a Transition Property for p* for γ > γc	596	(2)
Definition of P and Its Properties	598	(2)
The Feedback Generator Af and Its Preliminary Properties for γ > γc	600	(3)
The Operator P is a Solution of the Algebraic Riccati Equation, AREγy for γ > γc	603	(1)
The Semigroup Generated by (A -- BB*P) Is Uniformly Stable	604	(2)
The Case 0 < γ < γc: sup J0w,∞(y0) = +∞	606	(1)
Proof of Theorem 6.1.3.2	607	(1)
Part II: The Case Where eAT is Stable:	608	(1)
Motivation, Statement of Main Results	608	(4)
Minimization of J over u for w Fixed	612	(4)
Maximization of J0w(y0) over w: Existence of a Unique Optimal w*	616	(2)
Explicit Expressions of {u*, y*, w*} and P for γ > γc in Terms of the Data via E--1γ	618	(2)
Smoothing Properties of the Operators L, L*, W, W*: The Optimal u*, y*, w* Are Continuous in Time	620	(2)
A Transition Property for w* for γ > &gammxa;c	622	(4)
The Semigroup Property for y* for γ > γc and Its Stability	626	(1)
The Riccati Operator, P, for γ > γc	627	(3)
Optimal Control Problem with Nondefinite Quardratic Cost. The Stable, Analytic Case. A Brief Sketch	630
Notes on Chapter 6	639	(3)
References and Bibliography	642
Index