Lyapunov functionals and stability of stochastic functional differential equations

Cover......Page 1
Lyapunov Functionals and Stability of Stochastic Functional Differential Equations......Page 3
Preface......Page 5
Contents......Page 9
1.1.1 Description of Functional Differential Equations......Page 13
1.1.2 Reducing to Ordinary Differential Equations......Page 14
1.2 Method of Steps for Retarded Functional Differential Equations......Page 15
1.3 Characteristic Equation for Differential Equation with Discrete Delays......Page 17
1.4 The Influence of Small Delays on Stability......Page 33
1.5 Routh-Hurwitz Conditions......Page 37
2.1.1 Wiener Process and Its Numerical Simulation......Page 41
2.1.2 Itô Integral, Itô Stochastic Differential Equation, and Itô Formula......Page 43
2.2.1 Definitions of Stability and Basic Lyapunov-Type Theorems......Page 46
Step 1......Page 51
2.2.3 Auxiliary Lyapunov-Type Theorem......Page 52
2.3.1 Linear Stochastic Differential Equation......Page 55
2.3.2 System of Two Linear Stochastic Differential Equations......Page 59
2.3.3 Some Useful Inequalities......Page 62
2.4.1 Problem 1......Page 63
2.4.2 Problem 2......Page 64
3.1.1 The First Way of Constructing a Lyapunov Functional......Page 65
3.1.2 The Second Way of Constructing a Lyapunov Functional......Page 68
3.1.3 Some Particular Cases......Page 70
3.2.1 The First Way of Constructing a Lyapunov Functional......Page 74
3.2.2 The Second Way of Constructing a Lyapunov Functional......Page 75
3.2.3 The Third Way of Constructing a Lyapunov Functional......Page 76
3.2.4 The Fourth Way of Constructing a Lyapunov Functional......Page 77
3.2.5 One Generalization for Equation with n Delays......Page 79
3.3.1 Case n>1......Page 84
3.3.2 Some Particular Cases......Page 89
3.4.1 Equations with Variable Delays......Page 97
3.4.2 Equations with Variable Coefficients......Page 103
4.1 Linear Systems of Two Equations with Constant Delays......Page 109
4.2 Linear Systems of Two Equations with Distributed Delays......Page 114
4.3 Linear Systems of Two Equations with Variable Coefficients......Page 119
5.1.1 Scalar First-Order Differential Equation......Page 124
5.1.1.2 The Second Way of Constructing a Lyapunov Functional......Page 125
5.1.1.3 The Third Way of Constructing a Lyapunov Functional......Page 127
5.1.2 Scalar Second-Order Differential Equation......Page 131
5.2 Systems with Nonlinearities in Both Deterministic and Stochastic Parts......Page 135
5.3 Stability in Probability of Nonlinear Systems......Page 137
5.4 Systems with Fractional Nonlinearity......Page 142
5.4.1 Equilibrium Points......Page 143
5.4.2 Stochastic Perturbations, Centering, and Linearization......Page 145
5.4.3 Stability of Equilibrium Points......Page 146
5.4.4 Numerical Analysis......Page 152
6.1.1 One Delay in Deterministic Part and One Delay in Stochastic Part of Equation......Page 164
6.1.1.1 The First Way of Constructing a Lyapunov Functional......Page 165
6.1.1.2 The Second Way of Constructing a Lyapunov Functional......Page 166
6.1.2 Several Delays in Deterministic Part of Equation......Page 168
6.2 Distributed Delay......Page 171
6.3 Combination of Discrete and Distributed Delays......Page 175
6.4.1.1 The First Way of Constructing a Lyapunov Functional......Page 181
6.4.1.2 The Second Way of Constructing a Lyapunov Functional......Page 182
6.4.2 Several Delays in Deterministic Part of Equation......Page 185
6.5.1 The First Way of Constructing a Lyapunov Functional......Page 190
6.5.2 The Second Way of Constructing a Lyapunov Functional......Page 191
6.6 Equations with Unbounded Delays......Page 193
7.1 The Statement of the Problem......Page 197
7.2 Stability Theorems......Page 199
7.3 Application to Markov Chain with Two States......Page 203
7.3.2 The Second Stability Condition......Page 204
7.4 Numerical Simulation of Systems with Markovian Switching......Page 208
7.4.1 System Without Stochastic Perturbations......Page 209
7.4.2 System with Stochastic Perturbations......Page 212
7.4.3 System with Random Delay......Page 213
7.4.4 Some Generalization of Algorithm of Markov Chain Numerical Simulation......Page 216
8.1.1 Stabilization by the Control Depending on Trajectory......Page 219
8.1.2 Some Examples......Page 224
8.1.3 About Stabilization by the Control Depending on Velocity or on Acceleration......Page 229
8.1.4 Stabilization by Stochastic Perturbations......Page 230
8.2.1.1 The Deterministic Case......Page 232
8.2.2 Nonzero Steady-State Solutions......Page 236
8.2.3 Stable, Unstable, and One-Sided Stable Points of Equilibrium......Page 238
8.3.1 Stability of the Trivial Solution and Limit Cycles......Page 239
8.3.2 Nonzero Steady-State Solutions of the Nonlinear Model......Page 244
8.3.3 Stabilization of the Controlled Inverted Pendulum Under Influence of Markovian Stochastic Perturbations......Page 250
9.1 Introduction......Page 261
9.2 Two Points of Equilibrium, Stochastic Perturbations, Centering, and Linearization......Page 262
9.3 Sufficient Conditions for Stability in Probability for Both Equilibrium Points......Page 263
9.4 Numerical Illustrations......Page 265
10.1 System Under Consideration......Page 267
10.2.1 Equilibrium Points......Page 270
10.2.2 Stochastic Perturbations and Centering......Page 271
10.2.3 Linearization......Page 273
10.3 Stability of Equilibrium Point......Page 274
10.3.1 First Way of Constructing a Lyapunov Functional......Page 275
10.3.2 Second Way of Constructing a Lyapunov Functional......Page 282
10.3.3 Stability of the Equilibrium Point of Ratio-Dependent Predator-Prey Model......Page 287
11.1 Problem Statement......Page 293
11.2 Stability in Probability of the Equilibrium Point E0=(bµ1-1,0,0)......Page 294
11.3 Stability in Probability of the Equilibrium Point E*=(S*,I*,R*)......Page 299
11.4 Numerical Simulation......Page 305
12.1.1 Description of the Model of Alcohol Consumption......Page 307
12.1.2 Normalization of the Initial Model......Page 309
12.1.3 Existence of an Equilibrium Point......Page 310
12.1.4 Stochastic Perturbations, Centralization, and Linearization......Page 311
12.1.5 Stability of the Equilibrium Point......Page 312
12.1.6 Numerical Simulation......Page 319
12.2 Mathematical Model of Social Obesity Epidemic......Page 320
12.2.1 Description of the Considered Model......Page 321
12.2.2 Existence of an Equilibrium Point......Page 322
12.2.3 Stochastic Perturbations, Centralization, and Linearization......Page 324
12.2.4 Stability of an Equilibrium Point......Page 326
12.2.5 Numerical Simulation......Page 331
References......Page 334
Index......Page 348