Exponential stabilization of a flexible structure with a local interior control and under the presence of a boundary infinite memory

• Published in: Mathematical methods in the applied sciences Vol. 46; no. 2; pp. 2955 - 2971
• Main Authors: Chentouf, Boumediène, Mansouri, Sabeur
• Format: Journal Article
• Language: English
• Published: Freiburg Wiley Subscription Services, Inc 30.01.2023
• Subjects: Boundary control; exponential stability; flexible structure; Flexible structures; infinite memory; Linear operators; local interior control; minimal state; Operators (mathematics); Partial differential equations; Stabilization
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.8681

In this paper, we deal with the interior stabilization problem of a flexible structure governed by a hyperbolic partial differential equation coupled to two ordinary differential equations. Contrary to the previous works on the system, the boundary control is subject to the presence of an infinite memory term. In order to deal with such a nonlocal term, the minimal state approach is invoked. Specifically, a localized interior control is proposed in order to compensate the infinite memory effect. Thereafter, reasonable assumptions on the memory kernel are evoked so that the closed‐loop system is shown to be well‐posed thanks to semigroups theory of linear operators. Furthermore, the resolvent method is used to establish the exponential stability of the system.