On the finite-time capture of a fast moving target

In this work, we propose a feedback control law that enforces capture of a moving target by a slower pursuer in finite time. It is well known that if this problem is cast as a pursuit‐evasion differential game, then the moving target can always avoid capture by taking advantage of its speed superiority, provided that both the target and the pursuer are employing feedback strategies in the sense of Isaacs. Thus, in order to have a well‐posed pursuit problem, additional assumptions are required so that the pursuer can enforce capture of the faster target in finite time provided that it emanates from a set of ‘favorable’ initial positions, which constitute its winning set. In particular, we assume that the target's velocity either is constant and perfectly known to the pursuer (perfect information case) or can be decomposed into a dominant component, which is constant and known to the pursuer, and a second component that is uncertain and unknown to the pursuer (imperfect information case). It turns out that in both cases, the winning sets of the pursuer are pointed convex cones which have a common apex and a common axis of symmetry but different opening angles. We subsequently propose continuous feedback laws that enforce finite‐time capture while the pursuer never exits its winning set before capture takes place, for both cases.