  The purpose of this article is to study a stochastic control problem on a
junction, with control at the junction point. The problem of control is
formulated in the weak sense, using a relaxed control, namely a control which
takes values in the space of probability measures on a compact set. We prove
first the compactness of the admissible rules and the dynamic programming
principle (DPP). We complete this article by giving a verification Theorem for
the value function of the problem, using some recent results on quasi linear
non degenerate PDE posed on a junction, with non linear Neumann boundary
condition at the junction point. An example is given, where the optimal control
at the junction point is solution of a convex quadratic optimization problem
with linear constraints.
