Stability of Vector Differential Delay Equations

Cover......Page 1
Stability of Vector Differential Delay Equations......Page 4
Contents......Page 6
Preface......Page 10
1.1 Banach and Hilbert spaces......Page 12
1.2 Examples of normed spaces......Page 14
1.3 Linear operators......Page 15
1.4 Ordered spaces and Banach lattices......Page 18
1.5 The abstract Gronwall lemma......Page 19
1.6 Integral inequalities......Page 21
1.7 Generalized norms......Page 22
1.8 Causal mappings......Page 23
1.9 Compact operators in a Hilbert space......Page 25
1.10 Regularized determinants......Page 27
1.11 Perturbations of determinants......Page 28
1.12 Matrix functions of bounded variations......Page 30
1.13 Comments......Page 35
2.1 Notations......Page 36
2.2.1 Classical representations......Page 37
2.2.2 Multiplicative representations of the resolvent......Page 39
2.3 Norm estimates for resolvents......Page 41
2.4 Spectrum perturbations......Page 43
2.5.2 Functions regular on the convex hull of the spectrum......Page 47
2.5.3 Proof of Theorem 2.5.2......Page 48
2.6 Absolute values of entries of matrix functions......Page 50
2.7.1 A bound for similarity constants of matrices......Page 53
2.7.2 Proof of Theorem 2.7.1......Page 54
2.7.3 Applications of Theorem 2.7.1......Page 56
2.7.4 Additional norm estimates for functions of diagonalizable matrices......Page 58
2.8.1 Lower bounds......Page 59
2.8.2 Perturbations of matrix exponentials......Page 60
2.8.3 Proof of Theorem 2.8.1......Page 61
2.9 Matrices with non-negative off-diagonals......Page 62
2.10 Comments......Page 63
3.1 Description of the problem......Page 64
3.2 Existence of solutions......Page 66
3.3 Fundamental solutions......Page 68
3.4 The generalized Bohl–Perron principle......Page 69
3.5 Lp-version of the Bohl–Perron principle......Page 71
3.6 Equations with infinite delays......Page 74
3.7 Proof of Theorem 3.6.1......Page 75
3.8 Equations with continuous infinite delay......Page 77
3.9 Comments......Page 79
4.1 Statement of the problem......Page 80
4.2 Application of the Laplace transform......Page 83
4.3 Norms of characteristic matrix functions......Page 85
4.4 Norms of fundamental solutions of time-invariant systems......Page 87
4.5 Systems with scalar delay-distributions......Page 93
4.6 Scalar first-order autonomous equations......Page 94
4.7 Systems with one distributed delay......Page 100
4.8 Estimates via determinants......Page 103
4.10 Comments......Page 105
5.1 Sums of moduli of characteristic values......Page 106
5.2 Identities for characteristic values......Page 110
5.3 Multiplicative representations of characteristic functions......Page 112
5.4 Perturbations of characteristic values......Page 114
5.5 Perturbations of characteristic determinants......Page 116
5.7 Convex functions of characteristic values......Page 120
5.8 Comments......Page 122
6.1 Equations “close” to ordinary differential ones......Page 123
6.2 Equations with small delays......Page 127
6.3 Nonautomomous systems “close” to autonomous ones......Page 130
6.4 Equations with constant coefficients and variable delays......Page 133
6.5 Proof of Theorem 6.4.1......Page 135
6.6 The fundamental solution of equation (4.1)......Page 137
6.7 Proof of Theorem 6.6.1......Page 139
6.8 Comments......Page 140
7.1 Preliminary results......Page 141
7.2 The main result......Page 143
7.3 Norm estimates for block matrices......Page 145
7.4 Equations with one distributed delay......Page 146
7.5 Applications of regularized determinants......Page 148
7.6 Comments......Page 151
8.1 Vector equations with oscillating coefficients......Page 153
8.2 Proof of Theorem 8.1.1......Page 157
8.3 Scalar equations with several delays......Page 159
8.4 Proof of Theorem 8.3.1......Page 164
8.5 Comments......Page 166
9.1 The “freezing” method......Page 167
9.2 Proof of Theorem 9.1.1......Page 169
9.3 Perturbations of certain ordinary differential equations......Page 171
9.4 Proof of Theorems 9.3.1......Page 172
9.5 Comments......Page 174
10.1 Definitions and preliminaries......Page 175
10.2 Stability of quasilinear equations......Page 178
10.3 Absolute Lp-stability......Page 181
10.4 Mappings defined on Ω(ρ) ∩ 2......Page 183
10.5 Exponential stability......Page 185
10.6 Nonlinear equations “close” to ordinary differential ones......Page 187
10.7 Applications of the generalized norm......Page 189
10.8 Systems with positive fundamental solutions......Page 194
10.9 The Nicholson-type system......Page 196
10.10 Input-to-state stability of general systems......Page 199
10.11 Input-to-state stability of systems with one delay in linear parts......Page 200
10.12 Comments......Page 201
11.1 Preliminary results......Page 203
11.2 Absolute stability......Page 206
11.3 The Aizerman–Myshkis problem......Page 209
11.4 Proofs of Lemmas 11.3.2 and 11.3.4......Page 212
11.5 First-order nonlinear non-autonomous equations......Page 215
11.6 Comparison of Green’s functions to second-order equations......Page 218
11.7 Comments......Page 219
12.1 Introduction and statement of the main result......Page 221
12.2 Proof of Theorem 12.1.1......Page 223
12.3 Applications of matrix functions......Page 224
12.4 Comments......Page 226
13.1 Systems of semilinear equations......Page 227
13.2 Essentially nonlinear systems......Page 229
13.3 Nontrivial steady states......Page 231
13.4 Positive steady states......Page 232
13.6 Comments......Page 234
14.1 Preliminary results......Page 235
14.2 Volterra equations......Page 238
14.3 Differential delay equations......Page 239
14.4 Comments......Page 240
Appendix A The General Form of Causal Operators......Page 241
B.1 Definitions......Page 244
B.2 Properties of ..-Volterra operators......Page 245
B.3 Resolvents of ..-triangular operators......Page 247
B.4 Perturbations of block triangular matrices......Page 250
B.5 Block matrices close to triangular ones......Page 254
B.6 Diagonally dominant block matrices......Page 255
B.7 Examples......Page 257
Bibliography......Page 258
Index......Page 265