Engineering Optimization, 5th Edition

12 Optimal Control and Optimality Criteria Methods

12.1 Introduction

In this chapter we give a brief introduction to the following techniques of optimization:

1. Calculus of variations
2. Optimal control theory
3. Optimality criteria methods

If an optimization problem involves the minimization (or maximization) of a functional subject to the constraints of the same type, the decision variable will not be a number, but it will be a function. The calculus of variations can be used to solve this type of optimization problems. An optimization problem that is closely related to the calculus of variations problem is the optimal control problem. An optimal control problem involves two types of variables: the control and state variables, which are related to each other by a set of differential equations. Optimal control theory can be used for solving such problems. In some optimization problems, especially those related to structural design, the necessary conditions of optimality, for specialized design conditions, are used to develop efficient iterative techniques to find the optimum solution. Such techniques are known as optimality criteria methods.

12.2 Calculus of Variations

12.2.1 Introduction

The calculus of variations is concerned with the determination of extrema (maxima and minima) or stationary values of functionals. A functional can be defined as a function of several other functions. Hence the calculus of variations can be used to solve trajectory optimization problems. ...