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

Article type: Research Article

Authors: Bertrand, Pierre

Affiliations: Département de Mathématique, Université de Paris X, 200 Avenue de la République, 92000 Nanterre, France

Abstract: We give the development of jε: 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 Jε(υ,z)=$\frac{1}{2r}$‖z−zd‖2rL2r(0,T)+½N‖υ‖2L2(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(εk) for any k>0. We show that the boundary layer terms decay exponentially. We deduce, from the approximate optimality system, the expansion of jε, to order O(ε2k) and the associated control.

DOI: 10.3233/ASY-1989-2204

Journal: Asymptotic Analysis, vol. 2, no. 2, pp. 161-177, 1989