Boundary stabilization of non-diagonal systems by proportional feedback forms

Ionuţ Munteanu

doi:10.3934/cpaa.2021098

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2021, Volume 20, Issue 9: 3113-3128. Doi: 10.3934/cpaa.2021098

This issue Previous Article On the stability of boundary equilibria in Filippov systems Next Article Moderate deviation principles for unbounded additive functionals of distribution dependent SDEs

Boundary stabilization of non-diagonal systems by proportional feedback forms

Ionuţ Munteanu

Alexandru Ioan Cuza University, Department of Mathematics, and Octav Mayer Institute of Mathematics (Romanian Academy), Carol I No. 11, 700506 Iaşi, Romania

Received: December 31, 2020

Revised: April 30, 2021

Early access: June 2021

Published: September 2021

This work was supported by a grant of the Romanian Ministry of Research and Innovation, CNCS – UEFISCDI, project number PN-III-P1-1.1-TE-2019-0348, within PNCDI III

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Abstract

In this work, we are concerned with the problem of boundary exponential stabilization, in a Hilbert space $ H $, of parabolic type equations, namely equations for which their linear parts generate analytic $ C_0- $semigroups. We consider the case where the projection of the linear leading operator, on a given Riesz basis of $ H $, is non-diagonal. We do not assume that the linear operator has compact resolvent. Therefore, the Riesz basis is not necessarily an eigenbasis. The boundary stabilizer is given in a simple linear feedback form, of finite-dimensional structure, involving only the Riesz basis. To illustrate the results, at the end of the paper, we provide an example of stabilization of a fourth-order evolution equation on the half-axis.

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Mathematics Subject Classification: Primary: 93D15.

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