Judita Samuel

Numerical Solution of Differential Equations

We begin with the description of the simplest numerical method of solving differential equations. Although this method is not efficient and it is not used currently, it allows to understand the basic ideas of numerical procedures. At the same time, it is the starting point for more efficient methods. Although the results are valid for systems of differential equations of any order, we consider only the case of scalar equations, in order to avoid certain unessential difficulties.

General Problem of Optimal Control

The optimization problems studied in the preceeding chapters can be joined in a general framework. We shall describe this framework.

Control Systems. Stabilization of Linear Systems

The mathematical models for evolution processes that can be influenced through decisions are the dynamic control systems, defined by functions f : I × G × U ⊂ R × Rn × Rm → Rn; such system is written as x′ = f(t,x,u).

Some Unconstrained Dynamic Optimization Problems

The Calculus of Variations represents the oldest mathematical theory for optimization in the case of a controlled evolution.

Linear and Affine Differential Equations. Equations with Separated Variables

In order to understand the natural way in which differential equations occur in the study of evolutionary processes, we begin with some simple population growth models, that have also possible economic interpretations.

Linear-Quadratic Optimization on Finite Horizon

An optimal control problem, which can be viewed as a variant for stabilization, is the following.