3. Optimal Control Preliminaries

The preceding chapters focus on methods for solving NLP problems. In the remainder of the book, we turn our attention to the optimal control problem. An NLP problem is characterized by a finite set of variables x and constraints c. In contrast, optimal control problems can involve continuous functions such as y(t) and u(t). It will be convenient to view the optimal control problem as an infinite-dimensional extension of an NLP problem. However, practical methods for solving optimal control problems require Newton-based iterations with a finite set of variables and constraints. This goal can be achieved by transcribing or converting the infinite-dimensional problem into a finite-dimensional approximation.

Thus, the transcription method has three fundamental steps:

1. convert the dynamic system into a problem with a finite set of variables; then

2. solve the finite-dimensional problem using a parameter optimization method (i.e., the NLP subproblem); and then

3. assess the accuracy of the finite-dimensional approximation and if necessary repeat the transcription and optimization steps.

We will begin the discussion by focusing on the first step in the process, namely identifying the NLP variables, constraints, and objective function for common applications. In simple terms, we will focus on how to convert an optimal control problem into an NLP problem.

A dynamic system is usually characterized mathematically by a set of ordinary differential equations (ODEs).