Part I Retarded Type Systems
1 General Theory ............................................... 3
1.1 Preliminaries ........................................... 3
1.1.1 Initial Value Problem ............................ 3
1.1.2 Solutions ........................................ 4
1.1.3 State Concept .................................... 5
1.2 Existence and Uniqueness Issues ......................... 5
1.3 Continuity Properties .................................. 11
1.4 Stability Concepts ..................................... 13
1.5 Lyapunov-Krasovskii Approach ........................... 14
1.6 Notes and References ................................... 26
2 Single Delay Case ........................................... 27
2.1 Preliminaries .......................................... 28
2.1.1 Fundamental Matrix .............................. 28
2.1.2 Cauchy Formula .................................. 29
2.2 Exponential Stability .................................. 30
2.3 Problem Formulation .................................... 31
2.4 Delay-Free Case ........................................ 32
2.5 Computation of ν0(φ) ................................... 33
2.6 Lyapunov Matrices: Basic Properties .................... 37
2.7 Lyapunov Matrices: Limit Case .......................... 43
2.8 Lyapunov Matrices: New Definition ...................... 44
2.9 Lyapunov Matrices: Existence and Uniqueness Issues ..... 47
2.10 Lyapunov Matrices: Computational Issue ................. 56
2.11 Complete Type Functionals .............................. 58
2.12 Applications ........................................... 62
2.12.1 Quadratic Performance Index ..................... 62
2.12.2 Exponential Estimates ........................... 63
2.12.3 Critical Values of Delay ........................ 68
2.12.4 Robustness Bounds ............................... 69
2.13 Notes and References ................................... 73
3 Multiple Delay Case ......................................... 75
3.1 Preliminaries .......................................... 75
3.2 Quadratic Functionals .................................. 77
3.3 Lyapunov Matrices ...................................... 93
3.3.1 Exponentially Stable Case ....................... 93
3.3.2 General Case .................................... 94
3.4 Computational Schemes ................................. 106
3.4.1 Semianalytic Method ............................ 106
3.4.2 Scalar Equation ................................ 111
3.4.3 Numerical Scheme ............................... 113
3.4.4 Error Estimation ............................... 116
3.5 Exponential Estimates ................................. 122
3.6 Robustness Bounds ..................................... 124
3.6.1 Robust Stability Conditions: General Case ...... 124
3.6.2 Robust Stability Conditions: Scalar Case ....... 125
3.7 Applications .......................................... 127
3.7.1 Critical Values ................................ 127
3.7.2 The H2 Norm of a Transfer Matrix ............... 130
3.8 Notes and References .................................. 131
4 Systems with Distributed Delay ............................. 133
4.1 System Description .................................... 133
4.2 Quadratic Functionals ................................. 133
4.3 Lyapunov Matrices: Existence Issue .................... 141
4.4 Lyapunov Matrices: Uniqueness Issue ................... 145
4.5 Lyapunov Matrices: Computational Issue ................ 151
4.5.1 A Particular Case .............................. 151
4.5.2 A Special Case ................................. 159
4.5.3 Numerical Scheme ............................... 162
4.6 Complete Type Functionals ............................. 164
4.7 Exponential Estimates ................................. 168
Part II Neutral Type Systems
5 General Theory ............................................. 173
5.1 System Description .................................... 173
5.2 Existence Issue ....................................... 175
5.3 Continuity of Solutions ............................... 181
5.4 Stability Concepts .................................... 184
5.5 Lyapunov-Krasovskii Approach .......................... 185
5.6 Notes and References .................................. 200
6 Linear Systems ............................................. 201
6.1 Preliminaries ......................................... 201
6.1.1 Fundamental Matrix ............................. 202
6.1.2 Cauchy Formula ................................. 203
6.2 Lyapunov Matrices: Stable Case ........................ 204
6.3 Functional ν0(φ) ...................................... 211
6.4 Lyapunov Matrices: General Case ....................... 212
6.5 Existence and Uniqueness of Lyapunov Matrices ......... 220
6.6 Computational Issue ................................... 236
6.7 Spectral Properties ................................... 237
6.8 A New Form for Lyapunov Functionals ................... 239
6.9 Complete Type Functionals ............................. 243
6.10 Quadratic Bounds ...................................... 244
6.11 Applications .......................................... 249
6.11.1 Exponential Estimates .......................... 249
6.11.2 Quadratic Performance Index .................... 252
6.11.3 Robustness Bounds .............................. 253
6.12 Notes and References .................................. 254
7 Distributed Delay Case ..................................... 255
7.1 Preliminaries ......................................... 255
7.1.1 Fundamental Matrix ............................. 256
7.1.2 Cauchy Formula ................................. 256
7.2 Lyapunov Functionals .................................. 257
7.3 Lyapunov Matrices ..................................... 258
7.4 Lyapunov Matrices: New Definition ..................... 260
7.5 Existence Issue ....................................... 273
7.6 Computation Issue ..................................... 281
7.6.1 A Particular Case .............................. 281
7.6.2 A Special Case ................................. 288
7.7 A New Form of Lyapunov Functionals .................... 291
7.8 Complete Type Functionals ............................. 296
7.9 Quadratic Bounds ...................................... 298
7.10 The H2 Norm of a Transfer Matrix ...................... 303
References ................................................. 305
Index ......................................................... 309
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