The Brachistochrone Problem and Modern Control Theory

The purpose of this paper is to show that modern control theory, both in the form of the “classical” ideas developed in the 1950s and 1960s, and in that of later, more recent methods such as the “nonsmooth,” “very nonsmooth” and “differentialgeometric” approaches, provides the best and mathematically most natural setting to do justice to Johann Bernoulli’s famous 1696 “brachistochrone problem.” (For the classical theory, especially the smooth version of the Pontryagin Maximum Principle, see, e.g., Pontryagin et al. [31], Lee and Markus [26], Berkovitz [3]; for the nonsmooth approach, and the version of the Maximum Principle for locally Lipschitz vector fields, cf., e.g., Clarke [12, 13], Clarke et al. [14]; for very nonsmooth versions of the Maximum Principle, see Sussmann [35, 36, 39, 40, 41, 42, 43, 44]; for the differential-geometric approach, cf., e.g., Jurdjevic [24], Isidori [19], Nijmeijer and van der Schaft [30], Jakubczyk and Respondek [20], Sussmann [37, 40].) We will make our case in favor of modern control theory in two main ways. First, we will look at four approaches to the brachistochrone problem, presenting them in chronological order, and comparing them. Second, we will look at several “variations on the theme of the brachistochrone,” that is, at several problems closely related to the one of Johann Bernoulli. The first of our two lines of inquiry will be pursued in sections 3, 4, 5, and 6, devoted, respectively, to