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E-raamat: Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach

(University of Houston), (Collège de France, Paris), (University of Houston)
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Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical ApproachSuurem pilt
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A thorough mathematical analysis of controllability problems with a detailed investigation of methods for solving them numerically.

This book investigates how a user or observer can influence the behavior of systems mathematically and computationally. A thorough mathematical analysis of controllability problems is combined with a detailed investigation of methods used to solve them numerically; these methods being validated by the results of numerical experiments. In the first part of the book, the authors discuss the mathematics and numerics relating to the controllability of systems modeled by linear and non-linear diffusion equations; Part two is dedicated to the controllability of vibrating systems, typical ones being those modeled by linear wave equations; and finally, part three covers flow control for systems governed by the Navier-Stokes equations modeling incompressible viscous flow. The book is accessible to graduate students in applied and computational mathematics, engineering and physics; it will also be of use to more advanced practitioners.

Arvustused

"The book definitely has the perfume of those that Lions wrote during his prolific career. My congratulations to his two coworkers for having completed this task that reminded incomplete when he passed away in 2001. This book definitely fills a gap in the existing in literature on control and numerics of PDS, and I am sure it will influence future research in this area." Enrique Zuazua, Mathematical Reviews

Muu info

A thorough mathematical analysis of controllability problems with a detailed investigation of methods for solving them numerically.
Preface xi
Introduction 1(1)
What it is all about?
1(1)
Motivation
2(1)
Topologies and numerical methods
3(1)
Choice of the control
4(1)
Relaxation of the controllability notion
4(1)
Various remarks
5(4)
Part I Diffusion Models
Distributed and pointwise control for linear diffusion equations
9(115)
First example
9(3)
Approximate controllability
12(2)
Formulation of the approximate controllability problem
14(1)
Dual problem
15(2)
Direct solution to the dual problem
17(2)
Penalty arguments
19(3)
L∞ cost functions and bang-bang controls
22(6)
Numerical methods
28(29)
Relaxation of controllability
57(5)
Pointwise control
62(34)
Further remarks (I): Additional constraints on the state function
96(16)
Further remarks (II): A bisection based memory saving method for the solution of time dependent control problems by adjoint equation based methodologies
112(5)
Further remarks (III): A brief introduction to Riccati equations based control methods
117(7)
Boundary control
124(107)
Dirichlet control (I): Formulation of the control problem
124(2)
Dirichlet control (II): Optimality conditions and dual formulations
126(2)
Dirichlet control (III): Iterative solution of the control problems
128(5)
Dirichlet control (IV): Approximation of the control problems
133(10)
Dirichlet control (V): Iterative solution of the fully discrete dual problem (2.124)
143(3)
Dirichlet control (VI): Numerical experiments
146(9)
Neumann control (I): Formulation of the control problems and synopsis
155(8)
Neumann control (II): Optimality conditions and dual formulations
163(13)
Neumann control (III): Conjugate gradient solution of the dual problem (2.192)
176(2)
Neumann control (IV): Iterative solution of the dual problem (2.208), (2.209)
178(1)
Neumann control of unstable parabolic systems: a numerical approach
178(45)
Closed-loop Neumann control of unstable parabolic systems via the Riccati equation approach
223(8)
Control of the Stokes system
231(12)
Generalities. Synopsis
231(1)
Formulation of the Stokes system. A fundamental controllability result
231(3)
Two approximate controllability problems
234(1)
Optimality conditions and dual problems
234(2)
Iterative solution of the control problem (3.19)
236(2)
Time discretization of the control problem (3.19)
238(1)
Numerical experiments
239(4)
Control of nonlinear diffusion systems
243(34)
Generalities. Synopsis
243(1)
Example of a noncontrollable nonlinear system
243(2)
Pointwise control of the viscous Burgers equation
245(14)
On the controllability and the stabilization of the Kuramoto-Sivashinsky equation in one space dimension
259(18)
Dynamic programming for linear diffusion equations
277(6)
Introduction. Synopsis
277(1)
Derivation of the Hamilton--Jacobi--Bellman equation
278(1)
Some remarks
279(4)
Part II Wave Models
Wave equations
283(49)
Wave equations: Dirichlet boundary control
283(2)
Approximate controllability
285(1)
Formulation of the approximate controllability problem
286(1)
Dual problems
287(1)
Direct solution of the dual problem
288(1)
Exact controllability and new functional spaces
289(2)
On the structure of space E
291(1)
Numerical methods for the Dirichlet boundary controllability of the wave equation
291(24)
Experimental validation of the filtering procedure of Section 6.8.7 via the solution of the test problem of Section 6.8.5
315(4)
Some references on alternative approximation methods
319(1)
Other boundary controls
320(8)
Distributed controls for wave equations
328(1)
Dynamic programming
329(3)
On the application of controllability methods to the solution of the Helmholtz equation at large wave numbers
332(24)
Introduction
332(1)
The Helmholtz equation and its equivalent wave problem
332(2)
Exact controllability methods for the calculation of time-periodic solutions to the wave equation
334(1)
Least-squares formulation of the problem (7.8)--(7.11)
334(2)
Calculation of J'
336(1)
Conjugate gradient solution of the least-squares problem (7.14)
337(3)
A finite element--finite difference implementation
340(1)
Numerical experiments
341(8)
Further comments. Description of a mixed formulation based variant of the controllability method
349(6)
A final comment
355(1)
Other wave and vibration problems. Coupled systems
356(15)
Generalities and further references
356(3)
Coupled Systems (I): a problem from thermo-elasticity
359(8)
Coupled systems (II): Other systems
367(4)
Part III Flow Control
Optimal control of systems modelled by the Navier--Stokes equations: Application to drag reduction
371(55)
Introduction. Synopsis
371(2)
Formulation of the control problem
373(4)
Time discretization of the control problem
377(2)
Full discretization of the control problem
379(5)
Gradient calculation
384(4)
A BFGS algorithm for solving the discrete control problem
388(1)
Validation of the flow simulator
389(5)
Active control by rotation
394(14)
Active control by blowing and suction
408(11)
Further comments on flow control and conclusion
419(7)
Epilogue 426(3)
Further Acknowledgements 429(1)
References 430(20)
Index of names 450(4)
Index of subjects 454
Roland Glowinski is Cullen Professor of Mathematics at the University of Houston and Emeritus Professor at Laboratoire J. L. Lions, University P. and M. Curie, Paris. He is a Member of the French National Academy of Sciences and in 2004 won the Von Karman prize from the Society for Industrial and Applied Mathematics. He has written over 300 research papers and this is his 3rd book. The late Jacques-Louis Lions was a Professor at College de France, Paris. During his distinguished career he was elected an honorary member of over twenty learned societies and academies world-wide and received honorary degrees from nineteen universities. He was awarded a large number of international prizes, including the SIAM T and Idalia Reid Prize in Mathematics, the Prize of Japan and the John Von Neumann Prize. By the time of his death in 2001, he had authored over 20 books. Jiwen He is Associate Professor of Mathematics at the University of Houston. His research interests are numerical analysis, computational fluid dynamics, and control theory.
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