Differential Equations, Discrete Systems and Control

Inhaltsverzeichnis

• 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.