A simple optimal control problem involving approximation by monotone functions

For the problem of minimizing a suitable type of integral functionalJ[y] in the class_k of real, monotone nonincreasing functionsy which are Lipschitzian on a compact interval [a, b] with Lipschitz constantk, there is presented an existence theorem and a characterization of minimizing functions as solutions, in the sense of Filippov, of associated differential equations whose members involve discontinuities. For the problem of minimizingJ[y] in the class of all real, monotone nonincreasing functions for whichJ[y] exists, there is established an existence theorem and proof that, under suitable hypotheses, a solution of this second problem is the limit of solutions of the aforementioned problem ask . For the particular case in whichJ[y] is the integral of 1/2[y –h(t)]², whereh(t) is measurable and bounded on [a, b], it is shown that the minimizing function forJ[y] in the class is the derivative almost everywhere of the least concave majorant of the functionH(t)=₀^th(s)ds, t [a,b].This research was supported by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-749-65.