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

## Abstract

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: 〈γx, w〉g2 where w ε L2(Ω), gεL2(γ) and γ is a continuous operator for σ<3/4:H2σ(Ω)→L2(Ω). The free system 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 boundary vectors, so that the corresponding feedback solutions decay in the uniform operator norm exponentially at t → ∞. Here, this stabilization peoblem is pushed further and made more precise, under the additional assumption that the original free system be self-adjoint: we show, in fact, that one can further restrict the boundary vectors, so that the corresponding feedback solutions have the following more precise desirable structural property (the same enjoyed by free stable systems): they can be expressed as an infinite linear combination of decaying exponentials. A semigroup approach is employed. Since structure of feedback solutions is sought, the analysis here is much more technical and vastly different from [L8], where only norm upper bound was the goal. © 1982 Nicola Zanichelli Editore.

## Publication Title

Annali di Matematica Pura ed Applicata

## Recommended Citation

Lasiecka, I., & Triggiani, R.
(1982). Structural assignment of neumann boundary feedback parabolic equations: the unbounded case in the feedback loop.* Annali di Matematica Pura ed Applicata**, 132* (1), 131-175.
https://doi.org/10.1007/BF01760979