  In this paper we study the existence and uniqueness of Nash equilibria
(solution to competition-wise problems, with several controls trying to reach
possibly different goals) associated to linear partial differential equations
and show that, in some cases, they are also the solution of suitable
single-objective optimization problems (i.e. cooperative-wise problems, where
all the controls cooperate to reach a common goal). We use cost functions
associated with a particular linear parabolic partial differential equations
and distributed controls, but the results are also valid for more general
linear differential equations (including elliptic and hyperbolic cases) and
controls (e.g. boundary controls, initial value controls,...).
