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The problem of proving the existence of continuous viscosity solutions tonon-coercive stationary Hamilton-Jacobi equations, and thus describing theasymptotic behavior of solutions to the corresponding evolutive equations, hasbeen 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 byproviding a systematic extension of the celebrated weak KAM and Aubry-Mathertheories to the case of control systems with nonholonomic constraints. In thisframework, we consider an optimal control problem with a state equation ofnonholonomic type, where admissible trajectories are solutions of a nonlinearstate equation that is linear in the control variables. This equation isassociated with a family of smooth vector fields that satisfy the Hormandercondition, which implies the controllability of the system. In this case, theHamiltonian function of the control problem is not coercive, so results forTonelli Hamiltonians cannot be applied. To overcome these obstacles, we developan intrinsic approach based on the metric properties of the geometry induced onthe state space by the sub-Riemannian structure. Consequently, unlike theclassical Tonelli theory, the proof of the main results follows a more rigidroadmap that begins with the construction of Man\'e's critical constant. Thisstrategy, when applied to the Tonelli setting, allows for weaker regularityassumptions on the data. Nevertheless, we recover all the main results of theAubry-Mather theory in weak form, such as horizontal differentiability on theAubry set and a continuous Mather graph property.

Aubry-Mather approach to optimal control problems with nonholonimic constraints