# Rendezvous on the Line with Different Speeds and Markers That Can Be Dropped at Chosen Time
## 3 Problem Formulation
\begin{cases}v_{1}\cdot t,&t\in[0,t_{1}]\\ v_{1}\cdot t_{1}+v_{2}\cdot(t-t_{1}),&t\in[t_{1},t_{2}]\\ v_{1}\cdot t_{1}+v_{2}\cdot(t_{2}-t_{1})+v_{3}\cdot(t-t_{2}),&t\in[t_{2},t_{3}] \\ v_{1}\cdot t_{1}+v_{2}\cdot(t_{2}-t_{1})+v_{3}\cdot(t_{3}-t_{2})+v_{4}\cdot(t-t _{3}),&t\in[t_{3},t_{4}]\end{cases}  \tag{7}\]  
\[g(t)= \begin{cases}d_{1}\cdot t,&t\in[0,t_{1}]\\ d_{1}\cdot t_{1}+d_{2}\cdot(t-t_{1}),&t\in[t_{1},t_{2}]\\ d_{1}\cdot t_{1}+d_{2}\cdot(t_{2}-t_{1})+d_{3}\cdot(t-t_{2}),&t\in[t_{2},t_{3}] \\ d_{1}\cdot t_{1}+d_{2}\cdot(t_{2}-t_{1})+d_{3}\cdot(t_{3}-t_{2})+d_{4}\cdot(t -t_{3}),&t\in[t_{3},t_{4}]\end{cases}  \tag{8}\]  
_where \(v_{i}\in[-v,v]\), \(d_{i}=\pm 1\) and \(t_{1}\leq t_{2}\leq t_{3}\leq t_{4}\) are the rendezvous times._  
Proposition 1 and Corollary 1 are not new and are constantly used in the literature, see for instance Chapter 17.1 of [26]. We stress that player I having the smallest speed bound may move at a slower speed than the maximal one. Indeed, we will observe that for \(v\) small, the optimal strategy for player I is to not move before \(t_{2}\). The \"wait for mummy\" strategy is then optimal for starting the game. We consider that player I has at their disposal a marker that may be left at a chosen time. The marker helps player II, who stops following the strategy after finding the marker and continues in the same direction at maximal speed until rendezvousing with player I. The same arguments as the ones in Proposition 1 and Corollary 1 or Proposition 3 of [23], which show that player I move at a constant velocity before and after dropping the marker. There are four different cases to consider for the formulation of the problem depending on which interval, \([0,t_{1}],[t_{1},t_{2}],[t_{2},t_{3}],[t_{3},t_{4}]\) player I drops off the marker at. This leads to the following proposition that characterizes the optimal strategies. **Corollary 2**. : _When player I has a marker that can be dropped off at chosen time \(z\), the set of optimal strategies \(f\) for player I are given by_  
\[f(t)= \begin{cases}v_{0}\cdot t,&t\in[0,z]\\ v_{0}\cdot z+v_{1}\cdot(t-z),&t\in[z,t_{1}]\\ v_{0}\cdot z+v_{1}\cdot(t_{1}-z)+v_{2}\cdot(t-t_{1}),&t\in[t_{1},t_{2}]\\ v_{0}\cdot z+v_{1}\cdot(t_{1}-z)+v_{2}\cdot(t_{2}-t_{1})+v_{3}\cdot(t-t_{2}),&t \in[t_{2},t_{3}]\\ v_{0}\cdot z+v_{1}\cdot(t_{1}-z)+v_{2}\cdot(t_{2}-t_{1})+v_{3}\cdot(t_{3}-t_{2}) +v_{4}\cdot(t-t_{3}),&t\in[t_{3},t_{4}]\end{cases}  \tag{9}\]\[f(t)= \begin{cases}v_{1}\cdot t,&t\in[0,t_{1}]\\ v_{1}\cdot t_{1}+v_{0}\cdot(t-t_{1}),&t\in[t_{1},z]\\ v_{1}\cdot t_{1}+v_{0}\cdot(z-t_{1})+v_{2}\cdot(t-z),&t\in[z,t_{2}]\\ v_{1}\cdot t_{1}+v_{0}\cdot(z-t_{1})+v_{2}\cdot(t_{2}-z)+v_{3}\cdot(t-t_{2}),&t \in[t_{2},t_{3}]\\ v_{1}\cdot t_{1}+v_{0}\cdot(z-t_{1})+v_{2}\cdot(t_{2}-z)+v_{3}\cdot(t_{3}-t_{2}) +v_{4}\cdot(t-t_{3}),&t\in[t_{3},t_{4}]\end{cases}  \tag{10}\] \[\qquad\qquad\text{if }z\in[t_{1},t_{2}].\]  
\[f(t)= \begin{cases}v_{1}\cdot t,&t\in[0,t_{1}]\\ v_{1}\cdot t_{1}+v_{2}\cdot(t-t_{1}),&t\in[t_{1},t_{2}]\\ v_{1}\cdot t_{1}+v_{2}\cdot(t_{2}-t_{1})+v_{0}\cdot(t-t_{2}),&t\in[t_{2},z]\\ v_{1}\cdot t_{1}+v_{2}\cdot(t_{2}-t_{1})+v_{0}\cdot(z-t_{2})+v_{3}\cdot(t-z),&t \in[z,t_{3}]\\ v_{1}\cdot t_{1}+v_{2}\cdot(t_{2}-t_{1})+v_{0}\cdot(z-t_{2})+v_{3}\cdot(t_{3}-z )+v_{4}\cdot(t-t_{3}),&t\in[t_{3},t_{4}]\end{cases}  \tag{11}\] \[\qquad\qquad\text{if }z\in[t_{2},t_{3}].\]  
\[f(t)= \begin{cases}v_{1}\cdot t,&t\in[0,t_{1}]\\ v_{1}\cdot t_{1}+v_{2}\cdot(t-t_{1}),&t\in[t_{1},t_{2}]\\ v_{1}\cdot t_{1}+v_{2}\cdot(t_{2}-t_{1})+v_{3}\cdot(t-t_{2}),&t\in[t_{2},t_{3}] \\ v_{1}\cdot t_{1}+v_{2}\cdot(t_{2}-t_{1})+v_{3}\cdot(t_{3}-t_{2})+v_{0}\cdot(t- t_{3}),&t\in[t_{3},z]\\ v_{1}\cdot t_{1}+v_{2}\cdot(t_{2}-t_{1})+v_{3}\cdot(t_{3}-t_{2})+v_{0}\cdot(z-t_{3}) +v_{4}\cdot(t-z),&t\in[z,t_{4}]\end{cases}  \tag{12}\]  
\[f(t)=