Asymptotic Calculus of Variations

Abstract

This chapter presents the asymptotic calculus of variations. For the case of problems of optimal control, if one writes the optimality system, one ends up with a problem of singular perturbation, either for partial differential equations, in the case of no constraints, or for variational inequalities. A direct verification is possible for Dirichlet's boundary conditions; however, for general boundary conditions, the best method is the Energy method of L. Tartar. The averaging principle is a particular case of a general conjecture for nonquadratic functional and with constraints. The problem of optimal control (without constraints) is to find inf Jɛ(v), v ɛ L2(Q); it admits a unique solution uɛ: inf Jɛ(v) = Jɛ(uɛ). The asymptotic expansion of in f Jɛ(v ) in this case seems to be an open question. The asymptotic expansion depends on the geometry and more precisely on the position of r with respect to the x1's direction.