The spectrum of delay-differential equations: numerical methods - KTH

More documents

Recommendations

Info

Lambert W. This formula is valid for scalar DDEs and single-delay DDEs for which the matrices are simultaneously triangularizable. The treated formula is expressed using a matrix version of the Lambert W function. Some works in the literature state this formula in incorrect generality. We state some conditions on the system matrices such that the formula holds. If the system matrices are simultaneously triangularizable the formula holds. We show by counter-example that it does not hold in general. Chapter 3 deals with exact stability conditions of the DDE. We find exact conditions on the delays such that the DDE has a purely imaginary eigenvalue. All those combinations of the delays such that there is a purely imaginary eigenvalue (called critical delays) are parameterized. That is, we construct a mapping consisting of computable expressions, from a simple mathematical object (here ([−π, π] m−1 , Z m )) onto the set of all subsets of the critical delays. For DDEs of arbitrary dimension we give a mapping containing an eigenvalue problem. For the scalar case, we find an explicit expression containing only trigonometric functions. For non-scalar problems, an evaluation of the map consists of solving a quadratic eigenvalue problem of squared dimension. The constructed eigenvalue problem is large, even for DDEs of moderate size. For that reason, we present some notes on how the computational cost for one evaluation of the map can be reduced. In particular, we show that the matrix-vector product corresponding to the companion linearization of the quadratic eigenvalue problem can be evaluated by solving a Lyapunov equation. Most of the results in the chapter on critical delays are derived for retarded DDEs as well as neutral DDEs with an arbitrary number of delays. We also propose a new interpretation of a class of methods for delay-dependent stability, known as matrix pencil methods. The new interpretation is based on a similarity with two-parameter eigenvalue problems. We show that the polynomial eigenvalue problems in the matrix pencil methods are the generalized eigenvalue forms of a generalization of the two-parameter eigenvalue problem. The last chapter, i.e., Chapter 4, is about perturbation results for nonlinear eigenvalue problems. We present some notes on results in eigenvalue perturbation theory which can be generalized to nonlinear eigenvalue problems. A sensitivity formula for the movement of the eigenvalues extends nicely to nonlinear eigenvalue problems. We introduce a fixed point form for the nonlinear eigenvalue problem, and show that some methods in the literature can be interpreted as setvalued fixed point iterations. The convergence order of these types of iterations 7
8 Chapter 1. Introduction can be determined from an expression containing the left and right eigenvectors. We identify that the nonlinear eigenvalue problem belongs to the class of setvalued fixed point problems in fixed point theory. By using some results available in this field we generalize parts of an important theorem in perturbation theory referred to as the Bauer-Fike theorem.
Page 1: The spectrum of delay-differential
Page 4 and 5: Acronyms DDE delay-differential equ
Page 6 and 7: 3.3.2 Plotting critical curves . .
Page 8 and 9: tion. The formula is not new. We cl
Page 10 and 11: Matrix-Version der Lambert W-Funkti
Page 12 and 13: Chapter 1 Introduction Many physica
Page 14 and 15: An illustrative example The main co
Page 16 and 17: Imag 50 0 −50 −4 −2 0 2 Real
Page 20 and 21: Chapter 2 Computing the spectrum 2.
Page 22 and 23: 2.1. Introduction 11 For instance,
Page 24 and 25: 2.1. Introduction 13 solved in each
Page 26 and 27: 2.2. Methods for DDEs 15 of the typ
Page 28 and 29: 2.2. Methods for DDEs 17 by definin
Page 30 and 31: 2.2. Methods for DDEs 19 J = Wk(J)e
Page 32 and 33: 2.2. Methods for DDEs 21 Proof: The
Page 34 and 35: 2.2. Methods for DDEs 23 We note th
Page 36 and 37: 2.2. Methods for DDEs 25 The soluti
Page 38 and 39: 2.2. Methods for DDEs 27 The constr
Page 40 and 41: 2.2. Methods for DDEs 29 −3τ 3τ
Page 42 and 43: 2.2. Methods for DDEs 31 The idea o
Page 44 and 45: 2.2. Methods for DDEs 33 where the
Page 46 and 47: 2.2. Methods for DDEs 35 For instan
Page 48 and 49: 2.2. Methods for DDEs 37 Chebyshev
Page 50 and 51: 2.2. Methods for DDEs 39 (in time)
Page 52 and 53: 2.2. Methods for DDEs 41 a delay-in
Page 54 and 55: 2.2. Methods for DDEs 43 Example 2.
Page 56 and 57: 2.2. Methods for DDEs 45 N = 3 N =
Page 58 and 59: 2.3. Methods for nonlinear eigenval
Page 68 and 69:
2.3. Methods for nonlinear eigenval
Page 70 and 71:
2.3. Methods for nonlinear eigenval
Page 72 and 73:
2.4. Numerical examples 61 π/(n +
Page 74 and 75:
2.4. Numerical examples 63 We made
Page 76 and 77:
2.4. Numerical examples 65 2.4.2 Ra
Page 78:
2.4. Numerical examples 67 ance 10
Page 81 and 82:
70 Chapter 3. Critical Delays where
Page 83 and 84:
72 Chapter 3. Critical Delays 3.1 I
Page 85 and 86:
74 Chapter 3. Critical Delays For r
Page 87 and 88:
76 Chapter 3. Critical Delays the s
Page 89 and 90:
78 Chapter 3. Critical Delays This
Page 91 and 92:
80 Chapter 3. Critical Delays line
Page 93 and 94:
82 Chapter 3. Critical Delays The m
Page 95 and 96:
84 Chapter 3. Critical Delays after
Page 97 and 98:
86 Chapter 3. Critical Delays for e
Page 99 and 100:
88 Chapter 3. Critical Delays a Mö
Page 101 and 102:
90 Chapter 3. Critical Delays form,
Page 103 and 104:
92 Chapter 3. Critical Delays We no
Page 105 and 106:
94 Chapter 3. Critical Delays two d
Page 107 and 108:
96 Chapter 3. Critical Delays a par
Page 109 and 110:
98 Chapter 3. Critical Delays The r
Page 111 and 112:
100 Chapter 3. Critical Delays valu
Page 113 and 114:
102 Chapter 3. Critical Delays h 2
Page 115 and 116:
104 Chapter 3. Critical Delays The
Page 117 and 118:
106 Chapter 3. Critical Delays Let
Page 119 and 120:
108 Chapter 3. Critical Delays is w
Page 121 and 122:
110 Chapter 3. Critical Delays Exam
Page 123 and 124:
112 Chapter 3. Critical Delays case
Page 125 and 126:
114 Chapter 3. Critical Delays asso
Page 127 and 128:
116 Chapter 3. Critical Delays Lemm
Page 129 and 130:
118 Chapter 3. Critical Delays Proo
Page 131 and 132:
120 Chapter 3. Critical Delays and
Page 133 and 134:
122 Chapter 3. Critical Delays and
Page 135 and 136:
124 Chapter 3. Critical Delays Henc
Page 137 and 138:
126 Chapter 3. Critical Delays wher
Page 139 and 140:
128 Chapter 3. Critical Delays more
Page 141 and 142:
130 Chapter 3. Critical Delays the
Page 143 and 144:
132 Chapter 3. Critical Delays part
Page 145 and 146:
134 Chapter 3. Critical Delays b)
Page 147 and 148:
136 Chapter 3. Critical Delays whic
Page 149 and 150:
138 Chapter 3. Critical Delays for
Page 151 and 152:
140 Chapter 3. Critical Delays The
Page 153 and 154:
142 Chapter 3. Critical Delays Clea
Page 155 and 156:
144 Chapter 3. Critical Delays prob
Page 157 and 158:
146 Chapter 4. Perturbation of nonl
Page 159 and 160:
Page 161 and 162:
Page 163 and 164:
Page 165 and 166:
Page 167 and 168:
Page 169 and 170:
Page 171 and 172:
Page 173 and 174:
Page 175 and 176:
Page 177 and 178:
Page 179 and 180:
Page 181 and 182:
Page 183 and 184:
172 Appendix A. Appendix by ⎛ ⎜
Page 185 and 186:
174 BIBLIOGRAPHY [Bau85] H. Baumgä
Page 187 and 188:
176 BIBLIOGRAPHY [Bre06] , Solution
Page 189 and 190:
178 BIBLIOGRAPHY [Dat78] R. Datko,
Page 191 and 192:
180 BIBLIOGRAPHY [GMO + 06] M. Gu,
Page 193 and 194:
182 BIBLIOGRAPHY [HS05] P. Hövel a
Page 195 and 196:
184 BIBLIOGRAPHY [Lam91] J. Lambert
Page 197 and 198:
186 BIBLIOGRAPHY [MGWN06] W. Michie
Page 199 and 200:
188 BIBLIOGRAPHY [Nad69] S. B. Nadl
Page 201 and 202:
190 BIBLIOGRAPHY [Pio07] M. Piotrow
Page 203 and 204:
192 BIBLIOGRAPHY [SO06c] , A unique
Page 205 and 206:
194 BIBLIOGRAPHY [Vos06a] , Iterati
Page 207 and 208:
Index abstract Cauchy problem, 41,
Page 209 and 210:
198 INDEX Nyquist criterion, 83 off
Page 212 and 213:
Address: Institut Computational Mat
show all

The spectrum of delay-differential equations: numerical methods - KTH

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?