In this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such as methods of operator approximation with any given accuracy; operator interpolation techniques including a non-Lagrange interpolation; methods of system representation subject to constraints associated with concepts of causality, memory and stationarity; methods of system representation with an accuracy that is the best within a given class of models; methods of covariance matrix estimation;methods for low-rank matrix approximations; hybrid methods based on a combination of iterative procedures and best operator approximation; andmethods for information compression and filtering under condition that a filter model should satisfy restrictions associated with causality and different types of memory.As a result, the book represents a blend of new methods in general computational analysis,and specific, but also generic, techniques for study of systems theory ant its particularbranches, such as optimal filtering and information compression. - Best operator approximation,- Non-Lagrange interpolation,- Generic Karhunen-Loeve transform- Generalised low-rank matrix approximation- Optimal data compression- Optimal nonlinear filtering
Author(s): Richard Bellman
Publisher: Elsevier Science
Year: 1970
Language: English
Pages: 365
Contents......Page 16
Preface......Page 8
1.1. Introduction......Page 24
1.2. The First-order Linear Differential Equation......Page 25
1.3. Fundamental Inequality......Page 26
1.4. Second-order Linear Differential Equations......Page 28
1.5. Inhomogeneous Equation......Page 30
1.6. Lagrange Variation of Parameters......Page 31
1.7. Two-point Boundary Value Problem......Page 33
1.8. Connection with Calculus of Variations......Page 34
1.9. Green’s Functions......Page 35
1.10. Riccati Equation......Page 37
1.11. The Cauchy–Schwarz Inequality......Page 39
1.12. Perturbation and Stability Theory......Page 41
1.13. A Counter-example......Page 43
1.14. ∫∞ | f'(t)| dt < ∞......Page 44
1.15. ∫∞|f'(t)| dt < ∞......Page 45
1.1 6. Asymptotic Behavior......Page 46
1.17. The Equation u"– (1 + f (t))u = 0......Page 47
1.18. More Refined Asymptotic Behavior......Page 49
1.19. ∫∞ f2 dt < ∞......Page 50
1.20. The Second Solution......Page 52
1.21. The Liouville Transformation......Page 53
1.22. Elimination of Middle Term......Page 54
1.25. u” + (1 + f (t))u = 0; Asymptotic Behavior......Page 56
1.26. Asymptotic Series......Page 58
1.27. The Equation u’ = p(u, t)/q(u, t )......Page 60
1.28. Monotonicity of Rational Functions of u and t......Page 61
1.29. Asymptotic Behavior of Solutions of u’ = p(u, t)/q( u, t )......Page 62
Miscellaneous Exercises......Page 65
Bibliography and Comments......Page 74
2.1. Introduction......Page 77
2.2. Determinantal Solution......Page 78
2.3. Elimination......Page 81
2.4. Ill-conditioned Systems......Page 82
2.6. Vector Notation......Page 83
2.8. Vector Inner Product......Page 84
2.9. Matrix Notation......Page 86
2.10. Noncommutativity......Page 87
2.12. The Inverse Matrix......Page 88
2.13. Matrix Norm......Page 90
2.14. Relative Invariants......Page 91
2.15. Constrained Minimization......Page 94
2.16. Symmetric Matrices......Page 95
2.17. Quadratic Forms......Page 97
2.18. Multiple Characteristic Roots......Page 98
2.19. Maximization and Minimization of Quadratic Forms......Page 99
2.20. Min-Max Characterization of the λk......Page 100
2.21. Positive Definite Matrices......Page 102
2.22. Determinantal Criteria......Page 104
2.24. Canonical Representation for Arbitrary A......Page 105
2.25. Perturbation of Characteristic Frequencies......Page 107
2.26. Separation and Reduction of Dimensionality......Page 108
2.27. Ill-conditioned Matrices and Tychonov Regularization......Page 109
2.29. Positive Matrices......Page 111
2.30. Variational Characterization of λ(A)......Page 112
2.31. Proof of Minimum Property......Page 114
2.32. Equivalent Definition of λ(A)......Page 115
Miscellaneous Exercises......Page 117
Bibliography and Comments......Page 124
3.2. Vector-Matrix Calculus......Page 127
3.3. Existence and Uniqueness of Solution......Page 128
3.4. The Matrix Exponential......Page 130
3.5. Commutators......Page 131
3.6. Inhomogeneous Equation......Page 133
3.7. The Euler Solution......Page 134
3.8. Stability of Solution......Page 136
3.9. Linear Differential Equation with Variable Coefficients......Page 137
3.10. Linear Inhomogeneous Equation......Page 139
3.12. The Equation X' = AX + X B......Page 141
3.13. Periodic Matrices: the Floquet Representation......Page 143
3.14. Calculus of Variations......Page 144
3.15. Two-point Boundary Condition......Page 145
3.17. The Matrix Riccati Equation......Page 146
3.18. Kronecker Products and Sums......Page 147
3.19. AX + XB = C......Page 148
Miscellaneous Exercises......Page 150
Bibliography and Comments......Page 154
4.1. Introduction......Page 157
4.2. Dini-Hukuhara Theorem—I......Page 158
4.3. Dini-Hukuhara Theorem—II......Page 161
4.5. Existence and Uniqueness of Solution......Page 163
4.6. Poincaré-Lyapunov Stability Theory......Page 165
4.7. Proof of Theorem......Page 166
4.8. Asymptotic Behavior......Page 169
4.9. The Function φ(c)......Page 171
4.10. More Refined Asymptotic Behavior......Page 172
4.11. Analysis of Method of Successive Approximations......Page 173
4.13. Time-dependent Equations over Finite Intervals......Page 175
4.14. Alternative Norm......Page 178
4.1 5. Perturbation Techniques......Page 179
4.17. Solution of Linear Systems......Page 180
4.18. Origins of Two-point Boundary Value Problems......Page 181
4.19. Stability Theorem for Two-point Boundary Value Problem......Page 182
4.20. Asymptotic Behavior......Page 183
4.21. Numerical Aspects of Linear Two-point Boundary Value Problems......Page 184
4.22. Difference Methods......Page 186
4.24. Proof of Stability......Page 188
4.25. Analysis of Stability Proof......Page 189
4.27. Irregular Stability Problems......Page 191
4.28. The Emden–Fowler–Fermi–Thomas Equation......Page 193
Miscellaneous Exercises......Page 194
Bibliography and Comments......Page 205
5.1. Introduction......Page 210
5.2. Example of the Bubnov–Galerkin Method......Page 211
5.3. Validity of Method......Page 212
5.5. The General Approach......Page 213
5.6. Two Nonlinear Differential Equations......Page 215
5.7. The Nonlinear Spring......Page 216
5.9. Straightforward Perturbation......Page 219
5.11. The Van der Pol Equation......Page 221
5.13. The Linear Equation L(u) = g......Page 223
5.15. Nonlinear Case......Page 225
5.16. Newton–Raphson Method......Page 227
5.17. Multidimensional Newton–Raphson......Page 230
5.18. Choice of Initial Approximation......Page 231
5.19. Nonlinear Extrapolation and Acceleration of Convergence......Page 233
5.20. Alternatives to Newton–Raphson......Page 234
5.21. Lagrange Expansion......Page 235
5.22. Method of Moments Applied to Partial Differential Equations......Page 237
Miscellaneous Exercises......Page 238
Bibliography and Comments......Page 245
6.2. Differential Approximation......Page 248
6.4. Computational Aspects—I......Page 249
6.5. Computational Aspects—II......Page 250
6.6. Degree of Approximation......Page 251
6.7. Orthogonal Polynomials......Page 252
6.9. Extension of Classical Approximation Theory......Page 254
6.10. Riccati Approximation......Page 255
6.12. Application to Renewal Equation......Page 256
6.13. An Example......Page 259
6.14. Differential-Difference Equations......Page 261
6.15. An Example......Page 262
6.16. Functional-Differential Equations......Page 263
6.19. Mean-square Approximation......Page 265
6.20. Validity of the Method......Page 266
6.22. The Nonlinear Spring......Page 267
6.23. The Van der Pol Equation......Page 269
6.25. The Riccati Equation......Page 271
6.26. Higher-order Approximation......Page 273
6.27. Mean-square Approximation—Periodic Solutions......Page 274
Miscellaneous Exercises......Page 276
Bibliography and Comments......Page 278
7.2. The Euler Equation......Page 282
7.3. The Euler Equation and the Variational Problem......Page 283
7.4. Quadratic Functionals: Scalar Case......Page 284
7.5. Positive Definiteness for Small T......Page 286
7.6. Discussion......Page 287
7.8. Validity of the Method......Page 288
7.9. Monotone Behavior and Convergence......Page 290
7.10. Estimation of | u – v | in Terms of J(v) - J(u)......Page 291
7.11. Convergence of Coefficients......Page 292
7.12. Alternate Estimate......Page 293
7.13. Successive Approximations......Page 294
7.14. Determination of the Cofficients......Page 295
7.15. Multidimensional Case......Page 296
7.16. Reduction of Dimension......Page 297
7.17. Minimization of Inequalities......Page 298
7.18. Extension to Quadratic Functionals......Page 300
7.19. Linear Integral Equations......Page 302
7.20. Nonlinear Euler Equation......Page 303
7.21. Existence and Uniqueness......Page 304
7.23. Convexity and Uniqueness......Page 305
7.24. Implied Boundedness......Page 306
7.26. Functional Analysis......Page 307
7.27. The Euler Equation and Haar's Device......Page 309
7.28. Discussion......Page 310
7.30. Lagrange Multiplier......Page 311
7.32. Raising the Price Diminishes the Demand......Page 312
7.33. The Courant Parameter......Page 313
Miscellaneous Exercises......Page 314
Bibliography and Comments......Page 324
8.1. Equations Involving Parameters......Page 327
8.2. Stationary Values......Page 328
8.3. Characteristic Values and Functions......Page 329
8.4. Properties of Characteristic Values and Functions......Page 330
8.5. Generalized Fourier Expansion......Page 335
8.6. Discussion......Page 336
8.7. Rigorous Formulation of Variational Problem......Page 337
8.8. Rayleigh–Ritz Method......Page 338
8.10. Transplantation......Page 339
8.11. Positive Definiteness of Quadratic Functionals......Page 340
8.12. Finite Difference Approximations......Page 341
8.13. Monotonicity......Page 342
8.14. Positive Kernels......Page 343
Miscellaneous Exercises......Page 345
Bibliography and Comment......Page 352
Author Index......Page 354
Subject Index......Page 360
Mathematics in Science and Engineering......Page 364
