Introduction to Dynamic Systems

DAVID G. LUENBERGER is a professor in the School of Engineering at Stanford University. He has published four textbooks and over 70 technical papers. Professor Luenberger is a Fellow of the Institute of Electrical and Electronics Engineers and recipient of the 1990 Bode Lecture Award. His current research is mainly in investment science, economics, and planning.

• 1 Introduction1.1 Dynamic Phenomena 11.2 Multivariable Systems 21.3 A Catalog of Examples 41.4 The Stages of Dynamic System Analysis 102 Difference And Differential Equations2.1 Difference Equations 142.2 Existence and Uniqueness of Solutions 172.3 A First-Order Equation 192.4 Chain Letters and Amortization 212.5 The Cobweb Model 232.6 Linear Difference Equations 262.7 Linear Equations with Constant Coefficients 322.8 Differential Equations 382.9 Linear Differential Equations 402.10 Harmonic Motion and Beats 442.11 Problems 47Notes and References 543 Linear AlgebraAlgebraic Properties3.1 Fundamentals 563.2 Determinants 623.3 Inverses and the Fundamental Lemma 66Geometric Properties3.4 Vector Space 693.5 Transformations 733.6 Eigenvectors 773.7 Distinct Eigenvalues 803.8 Right and Left Eigenvectors 833.9 Multiple Eigenvalues 843.10 Problems 86Notes and References 894 Linear State Equations4.1 Systems Of First-Order Equations 904.2 Conversion to State Form 954.3 Dynamic Diagrams 974.4 Homogeneous Discrete-Time Systems 994.5 General Solution to Linear Discrete-Time Systems 1084.6 Homogeneous Continuous-Time Systems 1134.7 General Solution to Linear Continuous-Time Systems 1184.8 Embedded Statics 1214.9 Problems 124Notes and References 1305 Linear Systems With Constant Coefficients5.1 Geometric Sequences and Exponentials 1335.2 System Eigenvectors 1355.3 Diagonalization of a System 1365.4 Dynamics of Right and Left Eigenvectors 1425.5 Example: A Simple Migration Model 1445.6 Multiple Eigenvalues 1485.7 Equilibrium Points 1505.8 Example: Survival Theory in Culture 1525.9 Stability 1545.10 Oscillations 1605.11 Dominant Modes 1655.12 The Cohort Population Model 1705.13 The Surprising Solution to the Natchez Problem 1745.14 Problems 179Notes and References 1866 Positive Linear Systems6.1 Introduction 1886.2 Positive Matrices 1906.3 Positive Discrete-Time Systems 1956.4 Quality in a Hierarchy-The Peter Principle 1996.5 Continuous-Time Positive Systems 2046.6 Richardson's Theory of Arms Races 2066.7 Comparative Statics for Positive Systems 2116.8 Homans-Simon Model of Group Interaction 2156.9 Problems 217Notes and References 2227 Markov Chains7.1 Finite Markov Chains 2257.2 Regular Markov Chains and Limiting Distributions 2307.3 Classification of States 2357.4 Transient State Analysis 2397.5 Infinite Markov Chains 2457.6 Problems 248Notes and References 2538 Concepts Of Control8.1 Inputs, Outputs, and Interconnections 254Transform Methods8.2 z-Transforms 2558.3 Transform Solution of Difference Equations 2618.4 State Equations and Transforms 2668.5 Laplace Transforms 272State Space Methods8.6 Controllability 2768.7 Observability 2858.8 Canonical Forms 2898.9 Feedback 2968.10 Observers 3008.11 Problems 309Notes and References 3149 Analysis Of Nonlinear Systems9.1 Introduction 3169.2 Equilibrium Points 3209.3 Stability 3229.4 Linearization and Stability 3249.5 Example: The Principle of Competitive Exclusion 3289.6 Liapunov Functions 3329.7 Examples 3399.8 Invariant Sets 3459.9 A Linear Liapunov Function for Positive Systems 3479.10 An Integral Liapunov Function 3499.11 A Quadratic Liapunov Function for Linear Systems 3509.12 Combined Liapunov Functions 3539.13 General Summarizing Functions 3549.14 Problems 356Notes and References 36310 Some Important Dynamic Systems10.1 Energy in Mechanics 36510.2 Entropy in Thermodynamics 36710.3 Interacting Populations 37010.4 Epidemics 37610.5 Stability of Competitive Economic Equilibria 37810.6 Genetics 38210.7 Problems 389Notes and References 39111 Optimal Control11.1 The Basic Optimal Control Problem 39411.2 Examples 40111.3 Problems with Terminal Constraints 40511.4 Free Terminal Time Problems 40911.5 Linear Systems with Quadratic Cost 41311.6 Discrete-Time Problems 41611.7 Dynamic Programming 41911.8 Stability and Optimal Control 42511.9 Problems 427Notes and References 435References 436Index 441