Optimal Control and Monotone Smoothing Splines

Abstract

The solution to the problem of generating curves by driving the output of a particular, nilpotent single-input, single-output linear control system close to given waypoints is analyzed. The curves are furthermore constrained by an infinite dimensional, non-negativity constraint on one of the derivatives of the curve. The main theorem in this paper states that the optimal curve is a piecewise polynomial of known degree, and for the two-dimensional case, this problem is completely solved when the acceleration is controlled directly. The solution is obtained by exploiting a finite reparameterization of the problem, resulting in a dynamic programming formulation that can be solved analytically.