We show how any (generalized) supersolution of the Hamilton-Jacobi equation can be used to construct a feedback pursuit strategy which guarantees (to any given tolerance) a capture time not exceeding the solution’s value. If the supersolution is the value function, then a near-optimal pursuit strategy is obtained in this way. An important feature of the construction is its "universal" nature, i.e., the fact that the feedback law is uniformly effective on compact sets of initial conditions. This implies in particular that the feedback construction is one that exploits nonoptimal behavior on the part of the evader.