  The main purpose of this work is to obtain a comparison principle for
viscosity solutions of a system of elliptic Walsh's spider
Hamilton-Jacobi-Bellman equations, possessing a new boundary condition called
non linear local-time Kirchhoff's transmission. The main idea is to build test
functions at the neighborhood of the vertex solutions of ODE, with
well-designed coefficients. The key point is to impose a 'local-time'
derivative at the vertex absorbing the error term induced by - what we decide
to call here - the Kirchhoff's speed of the Hamiltonians.
