Control of singular problem via differentiation of a min-max

Abstract For Ω a smooth domain in R n with boundary Λ = Λ 0 ∪ Λ 1 , we are concerned with the wave equation y ″ − Δy = S in Q T =]0, T[ × Ω with = ∂/∂t , at source term satisfying S, S′ ϵ L 1 (0, T L 2 (Ω)) . A Dirichlet condition is imposed on Λ 0 and we consider an absorbing condition ∂y / ∂n + uy ′ = 0 in [0, T ] × gL 1 where u is the control.parameter. We introduce the cost function. J(u)= 1 2 ∫ T 0 ∫ Γ 1 γuy ′2 d Γ d t and using the Min-Max formulation of J we by-pasas the sensitivity analysis of u → y and obtain the gradient of J with a usual adjoint problem. We first present an abstract frame for this kind of problems. using the differentiability results of a Min-Max [1,2], which we very shortly deduce here, we show that the well posedness of the adjoint equation implies differentiability of the cost function governed by a linear well posed problem.