This is one of a handful of rendezvous problems where two players must find one another in a certain structured domain. See [AG2] for a thorough development of this subject. This is a symmetric rendezvous problem since each player is forced to adopt the same strategy. If we drop this constraint, Alpern and Gal [AG] have shown that the inf expected meeting time is 3.25.
Han, Du, Vera, and Zuluaga [HDVZ] have shown that strategies in which the players move at maximum speed and only change direction at integer times dominate among all possible strategies - thus reducing this problem to a discrete one. These same authors improve upon a series of results by tightening the upper and lower bounds, proving . Further, they conjecture .
Bibliography
*[A] S. Alpern, The rendezvous search problem. SIAM J. Control Optim. 33 (1995), no. 3, 673--683 MathSciNet
[AG1] S. Alpern and S. Gal, Rendezvous search on the line with distinguishable players. SIAM J. Control Optim. 33 (1995), no. 4, 1270--1276. MathSciNet
[AG2] S. Alpern and S. Gal, The theory of search games and rendezvous. International Series in Operations Research & Management Science, 55. Kluwer Academic Publishers, Boston, MA, 2003. MathSciNet
[HDVZ] Q. Han, D. Du, J. C. Vera, and L. F. Zuluaga, Improved bounds for the symmetric rendezvous search problem on the line
* indicates original appearance(s) of problem.