4.4.4 Fuller's problem

Consider again the double integrator (4.48) with the same control constraint $u\in[-1,1]$ . Let the cost functional be

$$J(u)=\int_{t_0}^{t_f}x_1^2(t)dt$$ (4.66)

and let the target set be $S=[t_0,\infty)\times\Big\{\Big({\textstyle{0}\atop \textstyle{0}}\Big)\Big\}$ . In other words, the final time is free and the final state is the origin, but this is not a minimal-time problem any more. The Hamiltonian is $H=p_1x_2+p_2u+p_0x_1^2$ and the optimal control must again satisfy (4.50), i.e., the switching function is $\varphi(t)=p_2^*(t)$ ; we can also see this from the general formula (4.60). The adjoint equation is

$$\begin{split}\dot p_1^*&=-2p_0^*x_1^*\\ \dot p_2^*&=-p_1^* \end{split}$$ (4.67)

We claim that singular arcs are ruled out. Indeed, along a singular arc $p_2^*$ must vanish. Inspecting the differential equations for $p_2^*$ , $p_1^*$ , and $x_1^*$ , we see that $p_1^*$ , $x_1^*$ , and $x_2^*$ must then vanish too. Thus the only possible singular arc is the trivial one consisting of the equilibrium at the origin.

It follows that all optimal controls are bang-bang, with switches occurring when $p_2^*$ equals 0. Our findings up to this point replicate those in the time-optimal setting of Section 4.4.1. There, we used the fact that $p_2^*$ depended linearly on $t$ to go further and show that the optimal control has at most one switch. The present adjoint equation (4.68) is different from (4.49) and so we can no longer reach the same conclusion. Instead, it turns out that the optimal solution has the following properties (which we state without proof):

1. Optimal controls are bang-bang with infinitely many switches.
2. Switching takes place on the curve $\Big\{\Big({\textstyle{x_1}\atop \textstyle{x_2}}\Big):x_1+\gamma\vert x_2\vert x_2=0\Big\}$ where $\gamma\approx 0.445$ .
3. Time intervals between consecutive switches decrease in geometric progression.

The last property is consistent with the fact that the final time must be finite (as the origin is reachable from every initial state in finite time, e.g., by using the time-optimal control). The occurrence of a switching pattern in which switching times form an infinite sequence accumulating near the final time is known as Fuller's phenomenon, or Zeno behavior. We note that the resulting control is measurable but not piecewise constant (see page ). Figure 4.17 shows the switching curve^4.5 and a sketch of an optimal state trajectory.

Figure: An optimal trajectory for Fuller's problem

We note that the switching curves in Fuller's problem and in the time-optimal control problem for the double integrator (treated in Section 4.4.1) are given by the same formula, but in the time-optimal problem we had a different value of $\gamma$ , namely, $\gamma=1/2$ . The nature of switching, however, is drastically different in the two cases. We can embed both problems in the parameterized family of problems with the cost functional $J(u)=\int_{t_0}^{t_f}\vert x_1(t)\vert^\nu dt$ where $\nu\ge 0$ is a parameter. For $\nu=0$ we recover the time-optimal problem while for $\nu=2$ we recover Fuller's problem. Interestingly, one can prove that there exists a ``bifurcation value" $\bar\nu\approx 0.35$ with the following property: for $\nu\in[0,\bar\nu]$ the optimal control is bang-bang with at most one switch, while for $\nu>\bar\nu$ we have Fuller's phenomenon (Zeno behavior).

The next exercise shows that Fuller's phenomenon is also observed in time-optimal control problems starting with dimension $3$ . Its solution can be based on Fuller's problem.