Comparison theorem for viscosity solutions of HJB equations posed on networks ramified with the nonlinear local-time Kirchhoff's boundary condition.

In this talk, I present the results of my last paper related to viscosity solutions for HJB equations posed on networks, of second order type and non degenerate at the vertex. The key point is to use the new boundary condition at the junction point, called: nonlinear local-time Kirchhoff's boundary condition, and build test functions with local-time derivatives absorbing the: Kirchhoff's speed of the Hamiltonians. Note that even without the presence of the external local-time variable in the HJB problems already studied in the literature, the ‘artificial’ introduction of this deterministic local-time variable, allowed an answer to a Lions-Souganidis problem to the fully nonlinear and non degenerate framework. Finally, it is important also to emphasize that the comparison theorem for viscosity solutions is stated here in the strong sense for the Kirchhoff’s boundary condition, namely without any dependency of the values of the Hamiltonians at the boundary, which is also innovative for Neumann problems in the nonlinear case for viscosity theory.