Hamilton-Jacobi equations for optimal control on junctions with unbounded running cost functions

This paper focuses on the viscosity solution approach to optimal control problems on junctions. Compared to Achdou et al. [Hamilton-Jacobi equations for optimal control on junctions and networks: ESAIM Control Optim. ESAIM Control Optim Calc Var. 2015;21(3):876-899] and Khang [Hamilton-Jacobi equations for optimal control on networks with entry or exit costs. ESAIM Control Optim Calc Var. 2018. (In press). EDP Sciences. ], we work on a less restrictive set of assumptions. We show that the value function is a unique viscosity solution of an associated Hamilton-Jacobi equation, and present some further properties of it. In addition, the viscosity solution method is used to establish a necessary and sufficient condition for an optimal control in a class of optimal control problems.