Abstract
We consider an optimal control on networks in the spirit of the works of Achdou et al. [NoDEA Nonlinear Differ. Equ. Appl. 20 (2013) 413–445] and Imbert et al. [ESAIM: COCV 19 (2013) 129–166]. 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. [ESAIM: COCV 21 (2015) 876–899] and one based on partial differential equations techniques inspired by a recent work of Lions and Souganidis [Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 27 (2016) 535–545].
Highlights
A network is a set of items, referred to as vertices or nodes, which are connected by edges
An optimal control problem is an optimization problem where an agent tries to minimize a cost which depends on the solution of a controlled ordinary differential equation (ODE)
The HJB equation has a unique viscosity solution characterizing by this way the value function
Summary
A network (or a graph) is a set of items, referred to as vertices or nodes, which are connected by edges (see Fig. 1 for example). The most important part of the paper will be devoted to two different proofs of a comparison principle leading to the well-poseness of (1.1): the first one uses arguments from optimal control theory coming from Barles et al [6, 7] and Achdou et al [3]; the second one is inspired by Lions and Souganidis [19] and uses arguments from the theory of PDEs. The paper is organized as follows: Section 2 deals with the optimal control problems with entry and exit costs: we give a simple example in which the value function is discontinuous at the vertex O, and prove results on the structure of the value function near O.
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