Professor Andrew Gard of the mathematics and computer science department at Lake Forest College, Illinois, US, has extensively explored such situations.
“The classical pursuit problem considers the path traced by a point in space as it charges directly toward a moving target. But what if the target has more lofty goals than mere escape? To what degree can it control the path taken by its pursuer? We prove that the pursuer can be led to any point in without being allowed to close more than an arbitrarily small distance en route.”
See : The Wild Goose Chase Problem in The American Mathematical Monthly 125(7):602-611 · August 2018
[Research research by Martin Gardiner]