Influence of a singular perturbation on the infimum of some functionals

AbstractLet Ω be a bounded, open subset of Rn. A class of problems in optimal control theory is considered in which constraints and a singular perturbation are involved. The state equation, depending on a real parameter ε ⩾ 0, is of the form εLz + g(z) = v (∗) where L is a self-adjoint operator in L2(Ω) and g is a non-linear function. The cost function is defined for some p ⩾ 1, N > 0 and zd ϵ L2p(Ω) by J(v, z) = ∝Ω{¦z−zd¦2p + N ¦v¦2)dx. The infimum of J(v, z) for ʋϵUad and (v, z) satisfying (∗) is denoted by jϵ for any ϵ ⩾ 0. •—If g is increasing and L⩾0 with (g, L) satisfying some natural conditions which insure well-posedness of (∗) for every ϵ > 0, then limϵ → 0(jϵ) = j0.•—If g is non-decreasing but remains constant on some interval, the above statement becomes false in general, even for L = −Δ with Dirichlet boundary conditions in Ω. Formulas are given in a typical case.•—If g is increasing and L < 0, in general limϵ → 0(jϵ) ≠ j0 The case where Ω = ]0, T[, D(L) = H2 ∩ H10(Ω), Lz = zn and p = 3, g(z) = z3 N > 0 is studied. A formula of ergodic type is established in order to specify the limit of (jϵ) when Uad={λ} and λ ≠ 0 is a real constant