Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach (Encyclopedia of Mathematics and its Applications (No. 117))

Cover......Page 1
Frontmatter......Page 2
Contents......Page 8
Preface......Page 12
I.1 What it is all about?......Page 14
I.2 Motivation......Page 15
I.3 Topologies and numerical methods......Page 16
I.5 Relaxation of the controllability notion......Page 17
I.6 Various remarks......Page 18
Part I - Diffusion Models......Page 20
1.1 First example......Page 22
1.2 Approximate controllability......Page 25
1.3 Formulation of the approximate controllability problem......Page 27
1.4 Dual problem......Page 28
1.5 Direct solution to the dual problem......Page 30
1.6 Penalty arguments......Page 32
1.7 L[INFINITY] cost functions and bang-bang controls......Page 35
1.8 Numerical methods......Page 41
1.9 Relaxation of controllability......Page 70
1.10 Point wise control......Page 75
1.11 Further remarks (I): Additional constraints on the state function......Page 109
1.12 Further remarks (II): A bisection based memory saving method for the solution of time dependent control problems by adjoint equation based methodologies......Page 125
1.13 Further remarks (III): A brief introduction to Riccati equations based control methods......Page 130
2.1 Dirichlet control (I): Formulation of the control problem......Page 137
2.2 Dirichlet control (II): Optimality conditions and dual formulations......Page 139
2.3 Dirichlet control (III): Iterative solution of the control problems......Page 141
2.4 Dirichlet control (IV): Approximation of the control problems......Page 146
2.5 Dirichlet control (V): Iterative solution of the fully discrete dual problem (2.124)......Page 156
2.6 Dirichlet control (VI): Numerical experiments......Page 159
2.7 Neumann control (I): Formulation of the control problems and synopsis......Page 168
2.8 Neumann control (II): Optimality conditions and dual formulations......Page 176
2.9 Neumann control (III): Conjugate gradient solution of the dual problem (2.192)......Page 189
2.11 Neumann control of unstable parabolic systems: a numerical approach......Page 191
2.12 Closed-loop Neumann control of unstable parabolic systems via the Riccati equation approach......Page 236
3.2 Formulation of the Stokes system. A fundamental controllability result......Page 244
3.4 Optimality conditions and dual problems......Page 247
3.5 Iterative solution of the control problem (3.19)......Page 249
3.6 Time discretization of the control problem (3.19)......Page 251
3.7 Numerical experiments......Page 252
4.2 Example of a noncontrollable nonlinear system......Page 256
4.3 Pointwise control of the viscous Burgers equation......Page 258
4.4 On the controllability and the stabilization of the Kuramoto-Sivashinsky equation in one space dimension......Page 272
5.1 Introduction. Synopsis......Page 290
5.2 Derivation of the Hamilton--Jacobi--Bellman equation......Page 291
5.3 Some remarks......Page 292
Part II - Wave Models......Page 294
6.1 Wave equations: Dirichlet boundary control......Page 296
6.2 Approximate controllability......Page 298
6.3 Formulation of the approximate controllability problem......Page 299
6.4 Dual problems......Page 300
6.5 Direct solution of the dual problem......Page 301
6.6 Exact controllability and new functional spaces......Page 302
6.8 Numerical methods for the Dirichlet boundary controllability of the wave equation......Page 304
6.9 Experimental validation of the filtering procedure of Section 6.8.7 via the solution of the test problem of Section 6.8.5......Page 328
6.10 Some references on alternative approximation methods......Page 332
6.11 Other boundary controls......Page 333
6.12 Distributed controls for wave equations......Page 341
6.13 Dynamic programming......Page 342
7.2 The Helmholtz equation and its equivalent wave problem......Page 345
7.4 Least-squares formulation of the problem (7.8)--(7.11)......Page 347
7.5 Calculation of J[PRIME]......Page 349
7.6 Conjugate gradient solution of the least-squares problem (7.14)......Page 350
7.7 A finite element--finite difference implementation......Page 353
7.8 Numerical experiments......Page 354
7.9 Further comments. Description of a mixed formulation based variant of the controllability method......Page 362
7.10 A final comment......Page 368
8.1 Generalities and further references......Page 369
8.2 Coupled Systems (I): a problem from thermo-elasticity......Page 372
8.3 Coupled systems (II): Other systems......Page 380
Part III - Flow Control......Page 382
9.1 Introduction. Synopsis......Page 384
9.2 Formulation of the control problem......Page 386
9.3 Time discretization of the control problem......Page 390
9.4 Full discretization of the control problem......Page 392
9.5 Gradient calculation......Page 397
9.6 A BFGS algorithm for solving the discrete control problem......Page 401
9.7 Validation of the flow simulator......Page 402
9.8 Active control by rotation......Page 407
9.9 Active control by blowing and suction......Page 421
9.10 Further comments on flow control and conclusion......Page 432
Epilogue......Page 439
Further Acknowledgements......Page 442
References......Page 443
Index of names......Page 463
Index of subjects......Page 467