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

Manh Khang Dao (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

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].

Mots clés: Optimal control; networks; Hamilton-Jacobi equation; viscosity solutions; uniqueness; switching cost

Domaines: Mathématiques [math]

Manh Khang Dao. Hamilton-Jacobi equations for optimal control on networks with entry or exit costs. ESAIM: Control, Optimisation and Calculus of Variations, 2019, 25, pp.15. ⟨10.1051/cocv/2018003⟩. ⟨hal-01548133v3⟩