Asymptotic expansion for the optimal cost of an ill-posed problem with singular perturbations

Abstract

We give the development of j_ε: the optimal cost of the nonlinear problem with singular perturbations with state equation

−εz″(t)−f(z(t))=υ and z(0)=z(T)=0

and with cost

J_ε(υ,z)= $\frac{1}{2r}$ ‖z−z_d‖^2r_L^2r(0,T)+½N‖υ‖²_L²(0,T).

We make a formal expansion of the optimality system. In the case without constraints, we introduce boundary layer terms to approximate it to order 0(ε^k) for any k>0. We show that the boundary layer terms decay exponentially. We deduce, from the approximate optimality system, the expansion of j_ε, to order O(ε^2k) and the associated control.