Inverse problems for linear parabolic equations using mixed formulations – Part 1: Theoretical analysis

Abstract

We consider the reconstruction of the solution of a parabolic equation posed in Ω × ( 0 , T ) , with a bounded open subset Ω of R N , from a partial distributed observation. We employ a least-squares technique and minimize the L 2 -norm of the distance from the observation to any solution. Taking the parabolic equation as the main constraint, the optimality conditions are reduced to a mixed formulation involving both the state to reconstruct and a Lagrange multiplier. The well-posedness of this mixed formulation, in particular the inf-sup property, is a consequence of classical energy estimates. We then reproduce the arguments to a linear first-order system, involving the normal flux, equivalent to the linear parabolic equation. The method, valid in any spatial dimension N, may also be employed to reconstruct solutions from boundary observations. With respect to the hyperbolic case, the parabolic situation requires, due to regularization properties, the introduction of an appropriate weight function so as to make the reconstruction stable with respect to standard Sobolev spaces.