Differential-Algebraic Equations: A Projector Based Analysis

Cover......Page 1
Differential-Algebraic Equations: A Projector Based Analysis......Page 4
Foreword by the Editors......Page 6
Preface......Page 8
Contents......Page 10
Notations......Page 16
Introduction......Page 19
1.1 Regular DAEs and the Weierstraß–Kronecker form......Page 28
1.2.1 Admissible matrix sequences and admissible projectors......Page 35
1.2.2 Decoupling by admissible projectors......Page 48
1.2.3 Complete decoupling......Page 55
1.2.4 Hierarchy of projector sequences for constant matrix pencils......Page 61
1.2.5 Compression to a generalized Weierstraß–Kronecker form......Page 62
1.2.6 Admissible projectors for matrix pairs in a generalized Weierstraß–Kronecker form......Page 66
1.3 Transformation invariance......Page 70
1.4 Characterizing matrix pencils by admissible projectors......Page 72
1.5 Properly stated leading term and solution space......Page 75
1.6 Notes and references......Page 77
2 Linear DAEs with variable coefficients......Page 82
2.1 Properly stated leading terms......Page 83
2.2.1 Basics......Page 85
2.2.2 Admissible projector functions and characteristic values......Page 90
2.2.3 Widely orthogonal projector functions......Page 100
2.3 Invariants under transformations and refactorizations......Page 104
2.4.1 Preliminary decoupling rearrangements......Page 111
2.4.2 Regularity and basic decoupling of regular DAEs......Page 115
2.4.3.1 Index-1 case......Page 129
2.4.3.2 Index-2 case......Page 131
2.4.3.3 General benefits from fine decouplings......Page 133
2.4.3.4 Existence of fine and complete decouplings......Page 136
2.5 Hierarchy of admissible projector function sequences for linear DAEs......Page 142
2.6 Fine regular DAEs......Page 143
2.6.1 Fundamental solution matrices......Page 144
2.6.2 Consistent initial values and flow structure......Page 148
2.6.3 Stability issues......Page 152
2.6.4 Characterizing admissible excitations and perturbation index......Page 157
2.7 Specifications for regular standard form DAEs......Page 162
2.8 The T-canonical form......Page 165
2.9 Regularity intervals and critical points......Page 171
2.10.1 Canonical forms......Page 185
2.10.2 Strangeness reduction......Page 189
2.10.3 Projector based reduction......Page 191
2.11.1 Measurable solutions......Page 196
2.11.2 Distributional solutions......Page 198
2.12 Notes and references......Page 199
3 Nonlinear DAEs......Page 207
3.1.1 Properly involved derivative......Page 208
3.1.2 Constraints and consistent initial values......Page 211
3.1.3 Linearization......Page 219
3.2 Admissible matrix function sequences and admissible projector functions......Page 222
3.3 Regularity regions......Page 232
3.4 Transformation invariance......Page 248
3.5 Hessenberg form DAEs of arbitrary size......Page 253
3.6 DAEs in circuit simulation......Page 263
3.7 Local solvability......Page 274
3.7.1 Index-1 DAEs......Page 275
3.7.2.1 Advanced decoupling of linear index-2 DAEs......Page 283
3.7.2.2 Nonlinear index-2 DAEs......Page 285
3.7.2.3 Index reduction step......Page 292
3.8 Advanced localization of regularity: including jet variables......Page 296
3.9 Operator settings......Page 305
3.9.1 Linear case......Page 307
3.9.2 Nonlinear case......Page 311
3.10 A glance at the standard approach via the derivative array and differentiation index......Page 314
3.11 Using structural peculiarities to ease models......Page 324
3.12 Regularity regions of DAEs with quasi-proper leading terms......Page 328
3.13 Notes and references......Page 331
4.1 Basic assumptions and notions......Page 339
4.2 Structure and solvability of index-1 DAEs......Page 342
4.3 Consistent initial values......Page 356
4.4 Notes and references......Page 358
5 Numerical integration......Page 360
5.1 Basic idea......Page 361
5.2.1 Backward differentiation formula......Page 366
5.2.2 Runge–Kutta method......Page 367
5.2.3 General linear method......Page 371
5.3.1 Backward differentiation formula......Page 373
5.3.2 Runge–Kutta method......Page 374
5.3.3 General linear method......Page 376
5.4 When do decoupling and discretization commute?......Page 377
5.5.1 Backward differentiation formula......Page 382
5.5.2 IRK(DAE) method......Page 385
5.5.3 General linear method......Page 390
5.6 Notes and references......Page 392
6.1 Preliminaries concerning explicit ODEs......Page 395
6.2 Contractive DAEs and B-stable Runge–Kutta methods......Page 398
6.3 Dissipativity......Page 404
6.4 Lyapunov stability......Page 407
6.5 Notes and references......Page 414
7 Computational linear algebra aspects......Page 416
7.1 Image and nullspace projectors......Page 417
7.2 Matters of a properly stated leading term......Page 419
7.3 The basic step of the sequence......Page 421
7.3.1 Basis representation methods......Page 423
7.3.2 Basis representation methods—Regular case......Page 425
7.3.3 Projector representation method......Page 426
7.4.1 Stepping level by level......Page 430
7.4.2 Involved version for the regular case......Page 432
7.4.3 Computing characteristic values and index check......Page 433
8.1 Practical index calculation......Page 435
8.2 Consistent initialization......Page 440
8.3 Numerical integration......Page 442
8.4 Notes and references......Page 453
9.1 Quasi-proper leading terms......Page 455
9.2 Quasi-admissible matrix function sequences and admissible projector functions......Page 460
9.3 Quasi-regularity......Page 466
9.4 Linearization......Page 469
9.5 A DAE transferable into SCF is quasi-regular......Page 471
9.6 Decoupling of quasi-regular linear DAEs......Page 476
9.7 Difficulties arising with the use of subnullspaces......Page 482
9.8 Notes and references......Page 485
9.9 Hierarchy of quasi-admissible projector function sequences for general nonlinear DAEs......Page 489
10 Nonregular DAEs......Page 491
10.1 The scope of interpretations......Page 492
10.2.1 Tractability index......Page 496
10.2.2 General decoupling......Page 500
10.2.2.1 G mu has full column rank......Page 503
10.2.2.2 Tractability index 1, G1 has a nontrivial nullspace......Page 507
10.2.2.3 Tractability index 2, G2 has a nontrivial nullspace......Page 511
10.3 Underdetermined nonlinear DAEs......Page 513
10.4 Notes and references......Page 516
11.1 Adjoint and self-adjoint DAEs......Page 518
11.2.1 A necessary extremal condition and the optimality DAE......Page 523
11.2.2 A particular sufficient extremal condition......Page 533
11.3 Specification for controlled DAEs......Page 535
11.4 Linear-quadratic optimal control and Riccati feedback solution......Page 538
11.4.1 Sufficient and necessary extremal conditions......Page 539
11.4.2 Riccati feedback solution......Page 540
11.5 Notes and references......Page 549
12 Abstract differential-algebraic equations......Page 552
12.1 Index considerations for ADAEs......Page 553
12.2 ADAE examples......Page 557
12.2.1.1 Wave equation......Page 558
12.2.1.2 Heat equation......Page 559
12.2.2 A semi-explicit system with parabolic and elliptic parts......Page 561
12.2.3 A coupled system of a PDE and Fredholm integral equations......Page 565
12.3 Linear ADAEs with monotone operators......Page 567
12.3.1 Basic functions and function spaces......Page 568
12.3.2 Galerkin approach......Page 571
12.3.3 Solvability......Page 577
12.3.4 Continuous dependence on the data......Page 584
12.3.5 Strong convergence of the Galerkin method......Page 587
12.4 Notes and references......Page 590
A.1 Projectors and subspaces......Page 594
A.2 Generalized inverses......Page 602
A.3 Parameter-dependent matrices and projectors......Page 604
A.4 Variable subspaces......Page 607
B.1 Proof of Lemma 2.12......Page 611
B.2 Proof of Lemma 2.41......Page 617
B.3 Admissible projectors for Nx\'+x=r......Page 624
C.1 A representation result......Page 638
C.2 ODEs......Page 640
C.3 Basics for evolution equations......Page 643
References......Page 648
Index......Page 657