Differential Equations, Discrete Systems and Control

1 Linear and Affine Differential Equations. Equations with Separated Variables.- 1.1 Differential Equations Modelling Growth Processes.- 1.2 Linear Differential Equations.- 1.3 Linear Affine Differential Equations.- 1.4 Simplest Models of Price Evolution in a Market Economy.- 1.5 Discrete - Time Models for Price Evolution.- 1.6 Simplest Models for Economic Growth.- 1.7 Discrete - Time Models for Economic Growth.- 1.8 Production Functions.- 1.9 Equations with Separated Variables.- 1.10 Notes and References.- 2 Linear Differential Equations with Constant Coefficients.- 2.1 Second Order Differential Equations with Constant Coefficients.- 2.2 Discrete - Time Second Order Linear Equations.- 2.3 Price Evolution in the Presence of Inventories.- 2.4 Economic Growth Models.- 2.5 Second Order Linear Affine Equations.- 2.6 The Phillips Model with Several Types of Autonomous Investment.- 2.7 Higher Order Linear Differential Equations with Constant Coefficients.- 2.8 Discrete - Time Linear Affine Equations.- 2.9 The Samuelson - Hicks Model for Economic Growth.- 2.10 Notes and References.- 3 Linear Systems with Constant Coefficients.- 3.1 General Form of Solutions.- 3.2 Matrix Exponential.- 3.3 Linear Affine Systems.- 3.4 Economic Models.- 3.5 Leontieff - type Models.- 3.6 Phase - Portrait for Second Order Linear Systems with Constant Coefficients.- 3.7 Notes and References.- 4 General Theory of Nonlinear Systems. Stability.- 4.1 Existence and Uniqueness Theorem for the Initial Value Problem.- 4.2 Equilibria. Stability. Continuous Time.- 4.3 Stability. Discrete Time.- 4.4 Discrete-Time Logistic Equation.- 4.5 Stable Polynomials.- 4.6 Some Properties of Matrices that occur in Economic Models.- 4.7 Notes and References.- 5 Numerical Solution of Differential Equations.-5.1 Euler Method.- 5.2 Richardson Extrapolation.- 5.3 Predictor - Corrector Methods.- 5.4 Numerical Quadrature.- 5.5 Adams Type Methods.- 5.6 Stiff Systems.- 5.7 Some Applications of Differential Equations in Numerical Analysis and Optimization.- 5.8 Notes and References.- 6 Control Systems. Stabilization of Linear Systems.- 6.1 Stabilization Problem. Stabilization by Linear State Feed-Back.- 6.2 Stabilization of Linear Systems by Using a Controller.- 6.3 Stabilization in an Economic Growth Model.- 6.4 A Monetary Policy Model.- 6.5 Stabilization of Discrete-Time Systems.- 6.6 A Discrete-Time Monetary Policy Model.- 6.7 Notes and References.- 7 Optimal Stabilization.- 7.1 Linear-Quadratic Optimization on Infinite Horizon. Continuous Time.- 7.2 Application to a Price Model.- 7.3 Optimal Stabilization in Discrete Time.- 7.4 Optimal Stabilization in a Discrete-Time Model of Price Evolution.- 7.5 Notes and References.- 8 Linear-Quadratic Optimization on Finite Horizon.- 8.1 Continuous Time.- 8.2 Applications.- 8.3 Discrete Time.- 8.4 Applications in Discrete Time.- 8.5 A Tracking Problem.- 8.6 A Simple Differential Game.- 8.7 Notes and References.- 9 Some Unconstrained Dynamic Optimization Problems.- 9.1 The Simplest Problem of the Calculus of Variations.- 9.2 A Macroeconomic Growth Model.- 9.3 A Discrete - Time Variational Problem.- 9.4 An Application.- 9.5 Unrestricted Optimal Control Problem in Discrete Time.- 9.6 An Application.- 9.7 Optimization with Linear Dynamics and Linear Cost. Continuous Time.- 9.8 Some Microeconomic Models.- 9.9 Optimization with Linear Dynamics and Linear Cost. Discrete Time.- 9.10 Applications.- 9.11 Notes and References.- 10 General Problem of Optimal Control.- 10.1 Problem Statement. General Theorems.- 10.2 Optimum CapitalAccumulation under the Minimum Time Objective.- 10.3 Reduction of Problems with Free Initial and Final Time to Problems on Fixed Horizon.- 10.4 An Abstract Multiplier Rule.- 10.5 Proof of Theorem 10.1.- 10.6 Notes and References.- References.