Dynamical Systems and Optimal Control. A Friendly Introduction

DYNAMICAL SYSTEMS AND OPTIMAL CONTROL......Page 1
Contents......Page 6
Preface......Page 10
1. Introduction to Modelling......Page 12
1.1.1 Malthus model......Page 13
1.1.2 Logistic models......Page 17
1.1.3 Phillips model......Page 21
1.1.5 Evolution of supply......Page 23
1.1.7 Lotka-Volterra predator-prey model......Page 25
1.1.8 Time-delay logistic equation......Page 27
1.2 Continuous Time and Discrete Time Models......Page 28
1.2.1 Differential and difference equations......Page 29
1.2.2 Systems of differential and difference equations......Page 32
2.1 Introduction......Page 36
2.2.1 Differential equations with separable variables......Page 37
2.2.2 Solow-Swan model......Page 40
2.2.3 Logistic model......Page 42
2.2.4 Linear equations......Page 43
2.2.5 Market dynamics......Page 48
2.2.6 Other types of equations......Page 49
2.3.1 Existence and uniqueness......Page 55
2.3.2 Maximal interval of existence......Page 59
2.4.1 Steady states, stability, phase space......Page 63
2.4.2 Stability by linearization......Page 67
2.5 A Neoclassical Growth Model......Page 68
2.6 Exercises......Page 72
3.2.1 Linear homogeneous equations......Page 76
3.2.2 Nonhomogeneous linear equations......Page 77
3.2.4 Cobweb model......Page 80
3.3.1 Orbits, stairstep diagram, steady states (fixed or equilibrium points)......Page 83
3.3.2 Steady states and stability......Page 85
3.3.3 Stability of periodic orbits......Page 89
3.3.4 Chaotic behavior......Page 93
3.3.5 Discrete logistic equation......Page 94
3.4 Exercises......Page 98
4.1.1 Homogeneous equations......Page 102
4.1.2 Nonhomogeneous equations......Page 106
4.2.1 Homogeneous equations......Page 108
4.2.2 Nonhomogeneous equations......Page 109
4.2.3 Stability......Page 112
4.2.4 Phillips model......Page 113
4.3 Exercises......Page 116
5.1.1 Homogeneous equations......Page 118
5.1.2 Fibonacci sequence......Page 121
5.1.3 Nonhomogeneous equations......Page 123
5.2.1 Homogeneous equations......Page 124
5.2.2 Nonhomogeneous equations......Page 125
5.2.3 Stability......Page 127
5.2.4 Accelerator model......Page 128
5.3 Exercises......Page 129
6.1 The Cauchy Problem......Page 130
6.2.2 Homogeneous systems......Page 132
6.2.3 Nonhomogeneous systems......Page 134
6.2.4 Equations of order n......Page 136
6.3.1 General integral......Page 137
6.4.1 Exponential matrix......Page 141
6.4.2 Cauchy problem and general integral......Page 147
6.4.3 Nonhomogeneous systems......Page 149
6.4.4 Stability of the zero solution......Page 150
6.5 Exercises......Page 151
7.1.1 Orbits......Page 154
7.1.2 Steady states, cycles and their stability......Page 158
7.1.3 Phase portrait......Page 160
7.2 Linear Systems. Classification of steady states......Page 165
7.3.1 The linearization method......Page 175
7.3.2 Outline of the Liapunov method......Page 178
7.4.1 Lotka-Volterra model......Page 181
7.4.2 A competitive equilibrium model......Page 187
7.5 Higher Dimensional Systems......Page 191
7.6 Exercises......Page 192
8.1 Linear Systems with Constant Coefficients......Page 194
8.1.1 Homogeneous systems......Page 195
8.1.2 Bidimensional homogeneous systems......Page 196
8.2 Stability......Page 200
8.2.1 Election polls......Page 202
8.2.2 A model of students partition......Page 204
8.3 Autonomous systems......Page 205
8.3.1 Discrete Lotka-Volterra model......Page 207
8.3.2 Logistic equation with delay......Page 208
8.4 Exercises......Page 210
9.1 Introduction......Page 214
9.2.1 Fixed boundaries. Euler equation......Page 217
9.2.2 Special cases of the Euler-Lagrange equation......Page 223
9.2.3 Free end values. Transversality conditions......Page 227
9.3 A Sufficient Condition of Optimality......Page 230
9.4 Infinite Horizon. Unbounded Interval. An Optimal Growth Problem......Page 231
9.5 The General Variation of a Functional......Page 235
9.6 Isoperimetric Problems......Page 240
9.7 Exercises......Page 245
10.1.1 Structure of a control problem. One-dimensional state and control......Page 250
10.1.2 Main questions and techniques......Page 253
10.2.1 Free final state. Necessary conditions......Page 254
10.2.2 Sufficient conditions......Page 258
10.2.3 Interpretation of the multiplier......Page 260
10.2.4 Maximum principle. Bounded controls......Page 261
10.2.5 Discounting. Current values......Page 262
10.2.6 Applications. Infinite horizon. Comparative analysis......Page 263
10.2.7 Terminal payoff and various endpoints conditions......Page 268
10.2.8 Discontinuous and bang-bang control. Singular solutions......Page 275
10.2.9 An advertising model control......Page 277
10.3.1 The simplest problem......Page 281
10.3.2 A discrete model for optimal growth......Page 284
10.3.3 State and control constraints......Page 285
10.3.4 Interpretation of the multiplier......Page 291
10.4 Exercises......Page 293
11.1 Introduction......Page 298
11.2 Continuous Time System. The Bellman Equation......Page 301
11.3 Infinite Horizon. Discounting......Page 306
11.4.1 The value function. The Bellman equation......Page 308
11.4.2 Optimal resource allocation......Page 312
11.4.3 Infinite horizon. Autonomous problems......Page 314
11.4.4 Renewable resource management......Page 315
A.1 Eigenvalues and Eigenvectors......Page 318
A.2 Functional Spaces......Page 321
A.3.1 Free optimization......Page 323
A.3.2 Constrained optimization. Equality constraints......Page 324
A.3.3 Constrained optimization. Inequality constraints......Page 326
References......Page 332
Subject index......Page 334