Structural assignment of neumann boundary feedback parabolic equations: the unbounded case in the feedback loop

Summary A parabolic equation defined on a bounded domain is considered, with input acting in the Neumann (or mixed) boundary conditions, and expressed as a specified feedback of the solution x of the form: x, wg₂ where w L₂(), gL₂() and is a continuous operator for <3/4:H ²()L ₂(). The freesystem is assumed unstable. In this case, the boundary feedback stabilization problem (in space dimension larger or equal to two) follows from an essentially more general result recently established by the authors in L8]:under algebraic (full rank), verifiable conditions at the unstable eigenvalues, one can select boundaryvectors, so that the corresponding feedback solutions decay in the uniform operator normexponentially at t .Here, this stabilization peoblem is pushed further and made more precise, under the additional assumption that the original freesystem be self-adjoint: we show, in fact, that one can further restrict the boundaryvectors, so that the corresponding feedback solutions have the following more precise desirable structural property (the same enjoyed by free stablesystems): they can be expressed as an infinite linear combination of decaying exponentials. A semigroup approach is employed. Since structureof feedback solutions is sought, the analysis here is much more technical and vastly different from L8],where only norm upper bound was the goal.The final version of this paper was completed while the authors were visiting the Istituto di Matematica « Pincherle » dell'Università di Bologna. Support from the Consiglio Nazionale delle Ricerche is gratefully acknowledged.