The process of transferring one state into another along a trajectory of a given differential system such that the time of transfer is minimal is known as the minimal-time problem, and it is one of the basic concerns of optimal control theory.

Minimal-time problems go back to the beginning of the calculus of variations. John Bernoulli's solution of the brachistochrone problem in 1697 was based on Fermat's principle of least time, which postulates that light traverses any medium in the least possible time. According to Goldstine's account (1980) of the history of the calculus of variations, Fermat announced that principle in 1662 in his collected works by saying that “nature operates by means and ways that are the easiest and fastest,” and he further differentiated that statement from the statement that “nature always acts along shortest paths” by citing an example from Galileo concerning the paths of particles moving under the action of gravity. Since then, time-optimal problems have remained important sources of inspiration during the growth of the calculus of variations.

In spite of the extensive literature on the subject, control theorists in the early 1950s believed that the classic theory did not adequately confront optimal problems that involved inequalities and was not applicable to problems of optimal control. Their early papers on time-optimal control problems paved the way to the maximum principle as a necessary condition for optimality.

This chapter begins with linear time-optimal problems and provides a selfcontained characterization of their time-optimal trajectories.