  The long-time average behavior of the value function in the calculus of
variations is known to be connected to the existence of the limit of the
corresponding Abel means. Still in the Tonelli case, such a limit is in turn
related to the existence of solutions of the critical Hamilton-Jacobi equation.
The goal of this paper is to address similar issues when set on the whole
Euclidean space and the Hamiltonian fails to be Tonelli. We first study the
convergence of the time-averaged value function as the time horizon goes to
infinity, proving the existence of the critical constant for a general control
system. Then, we show that the ergodic equation admits solutions for systems
associated with a family of vector fields which satisfies the Lie Algebra rank
condition. Finally, we construct a critical solution of the HJB equation on the
whole space which coincides with its Lax-Oleinik evolution.
