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CONTINUITY OF OPTIMAL CONTROL COSTS AND ITS APPLICATION TO WEAK KAM THEORY
 

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 Aubry-Mather 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 time-independent
Hamilton-Jacobi 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 Hamilton-Jacobi 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:

  

Source: Agrachev, Andrei - Functional Analysis Sector, Scuola Internazionale Superiore di Studi Avanzati (SISSA)

 

Collections: Engineering; Mathematics