  The problem of proving the existence of continuous viscosity solutions to
non-coercive stationary Hamilton-Jacobi equations, and thus describing the
asymptotic behavior of solutions to the corresponding evolutive equations, has
been open since the solution of the analogous question for the Tonelli case,
which was given by [LPV87]. In this work, we completely solve this problem by
providing a systematic extension of the celebrated weak KAM and Aubry-Mather
theories to the case of control systems with nonholonomic constraints. In this
framework, we consider an optimal control problem with a state equation of
nonholonomic type, where admissible trajectories are solutions of a nonlinear
state equation that is linear in the control variables. This equation is
associated with a family of smooth vector fields that satisfy the Hormander
condition, which implies the controllability of the system. In this case, the
Hamiltonian function of the control problem is not coercive, so results for
Tonelli Hamiltonians cannot be applied. To overcome these obstacles, we develop
an intrinsic approach based on the metric properties of the geometry induced on
the state space by the sub-Riemannian structure. Consequently, unlike the
classical Tonelli theory, the proof of the main results follows a more rigid
roadmap that begins with the construction of Man\'e's critical constant. This
strategy, when applied to the Tonelli setting, allows for weaker regularity
assumptions on the data. Nevertheless, we recover all the main results of the
Aubry-Mather theory in weak form, such as horizontal differentiability on the
Aubry set and a continuous Mather graph property.
