 
Summary: CONTINUITY OF OPTIMAL CONTROL COSTS AND ITS
APPLICATION TO WEAK KAM THEORY
ANDREI AGRACHEV AND PAUL W.Y. LEE
Abstract. We prove continuity of certain cost functions arising from optimal
control of affine control systems. We give sharp sufficient conditions for this con
tinuity. As an application, we prove a version of weak KAM theorem and consider
the AubryMather problems corresponding to these systems.
1. Introduction
Integrability of Hamiltonian systems has been a subject of considerable interest
for several decades. One way to understand the dynamics of such systems is to find
a family of smooth solutions, called generating functions, to the timeindependent
HamiltonJacobi equation. These generating functions define symplectic transfor
mations which transform the given completely integrable Hamiltonian system to a
much simpler one that are easily solvable.
On the contrary, if the Hamiltonian system is not completely integrable, then it
is natural to ask whether one can solve the HamiltonJacobi equation in certain
weak sense. This is accomplished in, what is known as, the weak KAM theorem
under certain assumptions on the Hamiltonian. More precisely, let L : TM R be
a Lagrangian defined on the tangent bundle TM of a compact manifold M which
satisfies the following conditions:
