Fuller’s Phenomena

Abstract

In his paper [F], Fuller studied the following optimal control problem. In the state space ℝ² with coordinates (x, y) he considers the control system: $\frac{{dx}}{{dt}} = u\quad \frac{{dy}}{{dt}} = x$ where the control u is restricted to the segment [-1, +1]. Given a point a in ℝ², he wants to determine the trajectories (x̂, ŷ, û) :[0, T̂] → ℝ² × [-1, +1] of the system, starting at a, ending at 0 and minimizing the cost $\frac{1}{2}\int_0^{\hat T} {\hat y{{(t)}^2}dt}$.