In this thesis, we propose a linear hybrid system describing heat flow on a medium composed by two rods connected by a point mass. We show that such a system can be obtained from a system describing heat flow of two rods connected by a thin wall of width 2є and density of 1/2є. By passing to a weak limit, we obtain the desired system.

We then show that the limiting system is null controllable with Dirichlet boundary control when the system's parameters satisfy a certain condition. Lastly, we consider simple parameters to show that the point mass system is null controllable with either Dirichlet or Neumann boundary control at one end.