Perfect splines and nonlinear optimal control theory

AbstractIn this paper we use a method from nonlinear optimal control theory to establish the “perfect spline” properties of a solution to a certain extremum problem. The problem is to minimize the L∞ norm of a nonlinear expression of the form F(t, x(t), ẋ(t), ẍ(t),…, x(n)(t)) over all sufficiently smooth functions x(t) which satisfy given boundary conditions. Under suitable assumptions, we show that a solution x0(t) must be such that F(t, x0(t), ẋ0(t),…, x0(n)(t)) is constant, and x0(n)(t) is piece-wise continuous with a finite number of jump discontinuities. This generalizes results by D. S. Carter, G. Glaeser, D. McClure, and others, who studied the same problem for linear differential expressions