On the value function of one-dimensional optimal control problems with junctions

We first study a one-dimensional optimal control problem with two controlled different dynamics in two half-lines. This may represent a two-edges node of a network on which we are controlling an evolution with different dynamics on different edges. By the point of view of the dynamic programming, and of the corresponding Hamilton-Jacobi-Bellman equation, the discontinuity of the dynamics at the node is in general a big mathematical problem. We introduce an approximation given by the use of a switching rule of the so-called delayed-relay type, and we study the passage to the limit when the parameter of the approximation goes to zero. Then we apply the same procedure to the case of a junction of three half-lines at one point (a three-edges node of a network).