Ricerc@Sapienza

We consider the Cauchy problem (Equation Presented) where - is a network and H is a positive homogeneous Hamiltonian which may change from edge to edge. In the first part of the paper, we prove that the Hopf-Lax type formula gives the (unique) viscosity solution of the problem. In the latter part of the paper we study a ame propagation model in a network and an optimal strategy to block a fire breaking up in some part of a pipeline; some numerical simulations are provided.

We prove a representation formula of Hopf-Lax type for solutions to Hamilton-Jacobi equation involving a Caputo time-fractional derivative. Equations of this type are associated with optimal control problems where the controlled dynamics is given by a time-changed stochastic process describing the trajectory of a particle subject to random trapping effects.