  We study a multi-objective variational problem of Herglotz' type with
cooperative linear coupling. We established the associated Euler-Lagrange
equations and the characteristic system for cooperative weakly coupled systems
of Hamilton-Jacobi equations. We also established the relation of the value
functions of this variational problem with the viscosity solutions of
cooperative weakly coupled systems of Hamilton-Jacobi equations. Comparing to
the previous work in stochastic frame, this approach affords a pure
deterministic explanation of this problem under more general conditions. We
also showed this approach is valid for general linearly coupling matrix for
short time.
