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 asoptimality 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. Afunctionalcan be defined as a function of
several other functions. Hence the calculus of variations can be used to solve trajectory
optimization problems.†The subject of calculus of variations is almost as old as the
calculusitself. The foundations of this subject were laid down by Bernoulli brothers and
later important contributions were made by Euler, Lagrange, Weirstrass, Hamilton, and
Bolzane. The calculus of variations is a powerful method for the solution of problems in
several fields, such as statics and dynamics of rigid bodies, general elasticity, vibrations,
optics, and optimization of orbits and controls. We shall see some of the fundamental
concepts of calculus of variations in this section.
†See Section 1.5 for the definition of a trajectory optimization problem.668 Engineering Optimization: Theory and Practice, Fourth Edition Singiresu S. Rao
Copyright © 2009 by John Wiley & Sons, Inc.