Hamilton-Jacobi equations for optimal control on networks with entry or exit costs

Abstract

We consider an optimal control on networks in the spirit of the works of Achdou et al. (2013) and Imbert et al. (2013). The main new feature is that there are entry (or exit) costs at the edges of the network leading to a possible discontinuous value function. We characterize the value function as the unique viscosity solution of a new Hamilton-Jacobi system. The uniqueness is a consequence of a comparison principle for which we give two different proofs, one with arguments from the theory of optimal control inspired by Achdou et al. (2014) and one based on partial differential equations techniques inspired by a recent work of Lions and Souganidis (2016).The first articles about optimal control problems in which the set of admissible states is a network (therefore the state variable is a continuous one) appeared in 2012: in [2], Achdou et al. derived the HJB equation associated to an infinite horizon optimal control on a network and proposed a suitable notion of viscosity solution. Obviously, the main difficulties arise at the vertices where the network does not have a regular differential structure. As a result, the new admissible test-functions whose restriction to each edge is C 1 are applied. Independently and at the same time, Imbert et al. [17] proposed an equivalent notion of viscosity solution for studying a Hamilton-Jacobi approach to junction problems and traffic flows. Both [2] and [17] contain first results on comparison principles which were improved later. It is also worth mentioning the work by Schieborn and Camilli [22], in which the authors focus on eikonal equations on networks and on a less general notion of viscosity solution. In the particular case of eikonal equations, Camilli and Marchi established in [10] the equivalence between the definitions given in [2,17,22].Since 2012, several proofs of comparison principles for HJB equations on networks, giving uniqueness of the solution, have been proposed. In [3], Achdou et al. give a proof of a comparison principle for a stationary HJB equationarising from an optimal control with infinite horizon, (therefore the Hamiltonian is convex) by mixing arguments from the theory of optimal control and PDE techniques. Such a proof was much inspired by works of Barles et al. [7,6], on regional optimal control problems in R d , (with discontinuous dynamics and costs). 2.A different and more general proof, using only arguments from the theory of PDEs was obtained by Imbert and Monneau in [16]. The proof works for quasi-convex Hamiltonians, and for stationary and time-dependent HJB equations. It relies on the construction of suitable vertex test functions. 3.A very simple and elegant proof, working for non convex Hamiltonians, has been very recently given by Lions and Souganidis [19,20].

Hamilton–Jacobi equations for optimal control on multidimensional junctions with entry costs

“…This feature is quite different from the one induced by the effect of entry costs in a network (i.e. when Γ is reduced to a point) considered in [14], where the value function at the junction point is a constant which is the minimum of the cost when the trajectory stays at the junction point forever and the cost when the trajectory enters immediately the edge that has the lowest possible cost.…”

“…Proof. The proof is classical and similar to the one in [14], so we skip it. Moreover, there exists ε > 0 such that V| P i \Γ is Lipschitz continuous in B(Γ, ε)∩P i \Γ.…”

“…For that later case, Camilli & Marchi [12] showed the equivalence between the definitions notion of viscosity solution given in [2,18] and [26]. Optimal control on networks with entry costs (and exit costs) has recently been considered by the first author [14].…”