Homicidal chauffeur problem

Problem

Two players move in the plane. The pursuer $P$ is a car: it moves with constant speed $w_1 > 0$ and has minimum turning radius $R > 0$. Its position $(x_1, y_1)$ and heading $\theta$ obey $$x_1' = w_1 \sin \theta, \qquad y_1' = w_1 \cos \theta, \qquad \theta' = \frac{w_1}{R} \phi, \qquad \phi(t) \in [-1, 1].$$ The evader $E$ moves with speed at most $w_2$, where $0 < w_2 < w_1$, in any direction: its position $(x_2, y_2)$ obeys $x_2' = w_2 \sin \psi$, $y_2' = w_2 \cos \psi$ with $\psi(t)$ free. Let $\ell > 0$ be the capture radius; capture happens at the first time the distance between $P$ and $E$ is at most $\ell$. The problem is a differential game between $P$ and $E$. It asks for which parameters $w_1, w_2, R, \ell$ and initial positions the pursuer can force capture in finite time against every play of the evader. It also asks for the optimal strategies when the pursuer minimises and the evader maximises the time to capture.