Multicriteria optimal control and vectorial Hamilton-Jacobi equation

In this paper we investigate a multicriteria optimal control problem associated to a preference relation based on the lexicographic order. We extend different notions of non-smooth analysis and control and show that the vector Value function is the unique vector lower semi-continuous solution to a suitable system of Hamilton-Jacobi equations in the sense of contingent solution or equivalently in the sense of extended viscosity solution.