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

, by Roland Glowinski , Jacques-Louis Lions , Jiwen He

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ISBN: 9780521885720 | 0521885728
Cover: Hardcover
Copyright: 4/21/2008

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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.

Preface	p. xi
Introduction	p. 1
What it is all about?	p. 1
Motivation	p. 2
Topologies and numerical methods	p. 3
Choice of the control	p. 4
Relaxation of the controllability notion	p. 4
Various remarks	p. 5
Diffusion Models
Distributed and pointwise control for linear diffusion equations	p. 9
First example	p. 9
Approximate controllability	p. 12
Formulation of the approximate controllability problem	p. 14
Dual problem	p. 15
Direct solution to the dual problem	p. 17
Penalty arguments	p. 19
L[superscript infinity] cost functions and bang-bang controls	p. 22
Numerical methods	p. 28
Relaxation of controllability	p. 57
Pointwise control	p. 62
Further remarks (I): Additional constraints on the state function	p. 96
Further remarks (II): A bisection based memory saving method for the solution of time dependent control problems by adjoint equation based methodologies	p. 112
Further remarks (III): A brief introduction to Riccati equations based control methods	p. 117
Boundary control	p. 124
Dirichlet control (I): Formulation of the control problem	p. 124
Dirichlet control (II): Optimality conditions and dual formulations	p. 126
Dirichlet control (III): Iterative solution of the control problems	p. 128
Dirichlet control (IV): Approximation of the control problems	p. 133
Dirichlet control (V): Iterative solution of the fully discrete dual problem (2.124)	p. 143
Dirichlet control (VI): Numerical experiments	p. 146
Neumann control (I): Formulation of the control problems and synopsis	p. 155
Neumann control (II): Optimality conditions and dual formulations	p. 163
Neumann control (III): Conjugate gradient solution of the dual problem (2.192)	p. 176
Neumann control (IV): Iterative solution of the dual problem (2.208), (2.209)	p. 178
Neumann control of unstable parabolic systems: a numerical approach	p. 178
Closed-loop Neumann control of unstable parabolic systems via the Riccati equation approach	p. 223
Control of the Stokes system	p. 231
Generalities. Synopsis	p. 231
Formulation of the Stokes system. A fundamental controllability result	p. 231
Two approximate controllability problems	p. 234
Optimality conditions and dual problems	p. 234
Iterative solution of the control problem (3.19)	p. 236
Time discretization of the control problem (3.19)	p. 238
Numerical experiments	p. 239
Control of nonlinear diffusion systems	p. 243
Generalities. Synopsis	p. 243
Example of a noncontrollable nonlinear system	p. 243
Pointwise control of the viscous Burgers equation	p. 245
On the controllability and the stabilization of the Kuramoto-Sivashinsky equation in one space dimension	p. 259
Dynamic programming for linear diffusion equations	p. 277
Introduction. Synopsis	p. 277
Derivation of the Hamilton-Jacobi-Bellman equation	p. 278
Some remarks	p. 279
Wave Models
Wave equations	p. 283
Wave equations: Dirichlet boundary control	p. 283
Approximate controllability	p. 285
Formulation of the approximate controllability problem	p. 286
Dual problems	p. 287
Direct solution of the dual problem	p. 288
Exact controllability and new functional spaces	p. 289
On the structure of space E	p. 291
Numerical methods for the Dirichlet boundary controllability of the wave equation	p. 291
Experimental validation of the filtering procedure of Section 6.8.7 via the solution of the test problem of Section 6.8.5	p. 315
Some references on alternative approximation methods	p. 319
Other boundary controls	p. 320
Distributed controls for wave equations	p. 328
Dynamic programming	p. 329
On the application of controllability methods to the solution of the Helmholtz equation at large wave numbers	p. 332
Introduction	p. 332
The Helmholtz equation and its equivalent wave problem	p. 332
Exact controllability methods for the calculation of time-periodic solutions to the wave equation	p. 334
Least-squares formulation of the problem (7.8)-(7.11)	p. 334
Calculation of J'	p. 336
Conjugate gradient solution of the least-squares problem (7.14)	p. 337
A finite element-finite difference implementation	p. 340
Numerical experiments	p. 341
Further comments. Description of a mixed formulation based variant of the controllability method	p. 349
A final comment	p. 355
Other wave and vibration problems. Coupled systems	p. 356
Generalities and further references	p. 356
Coupled Systems (I): a problem from thermo-elasticity	p. 359
Coupled systems (II): Other systems	p. 367
Flow Control
Optimal control of systems modelled by the Navier-Stokes equations: Application to drag reduction	p. 371
Introduction. Synopsis	p. 371
Formulation of the control problem	p. 373
Time discretization of the control problem	p. 377
Full discretization of the control problem	p. 379
Gradient calculation	p. 384
A BFGS algorithm for solving the discrete control problem	p. 388
Validation of the flow simulator	p. 389
Active control by rotation	p. 394
Active control by blowing and suction	p. 408
Further comments on flow control and conclusion	p. 419
Epilogue	p. 426
Further Acknowledgements	p. 429
References	p. 430
Index of names	p. 450
Index of subjects	p. 454
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Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach

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