{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T22:26:05Z","timestamp":1775082365032,"version":"3.50.1"},"reference-count":37,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,4,7]],"date-time":"2025-04-07T00:00:00Z","timestamp":1743984000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["12301579"],"award-info":[{"award-number":["12301579"]}]},{"name":"National Natural Science Foundation of China","award":["12301569"],"award-info":[{"award-number":["12301569"]}]},{"name":"National Natural Science Foundation of China","award":["3122019140"],"award-info":[{"award-number":["3122019140"]}]},{"name":"Special Fund for Basic Research Operating Costs of Central Universities","award":["12301579"],"award-info":[{"award-number":["12301579"]}]},{"name":"Special Fund for Basic Research Operating Costs of Central Universities","award":["12301569"],"award-info":[{"award-number":["12301569"]}]},{"name":"Special Fund for Basic Research Operating Costs of Central Universities","award":["3122019140"],"award-info":[{"award-number":["3122019140"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we investigate the stability of a 1-d wave equation with boundary difference-type delay control. We utilize the idea of system equivalence to find a system with known stability characteristics and select an appropriate regulation mechanism, ensuring that the original system becomes equivalent to the stable one. In this method, we adopt integral-type feedback control, utilizing integral kernel functions as parameters, and determine appropriate parameter functions. The specific steps are as follows: To begin with, an exponentially convergent system is selected as the desired target reference model. Next, we construct a bounded, reversible linear mapping to equate the studied system with the target model. During this process, we derive the expressions for the integral controller and the corresponding kernel function. Subsequently, we prove the solvability of the kernel function. By establishing equations for the kernel function and linear transformations, we find that the initial system exhibits equivalence to the desired model. Ultimately, based on the equivalence between the two systems, we conclude that the original system attains exponential stability under the integral-type feedback controller.<\/jats:p>","DOI":"10.3390\/axioms14040280","type":"journal-article","created":{"date-parts":[[2025,4,23]],"date-time":"2025-04-23T20:43:17Z","timestamp":1745440997000},"page":"280","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Exponential Stability of a Wave Equation with Boundary Delay Control"],"prefix":"10.3390","volume":"14","author":[{"given":"Yaru","family":"Xie","sequence":"first","affiliation":[{"name":"Department of Mathematics, Civil Aviation University of China, Tianjin 300300, China"}]},{"ORCID":"https:\/\/orcid.org\/0009-0004-3285-1544","authenticated-orcid":false,"given":"Congyue","family":"Tian","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Civil Aviation University of China, Tianjin 300300, China"}]},{"given":"Yanfang","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Henan Normal University, Xinxiang 453007, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,7]]},"reference":[{"key":"ref_1","first-page":"177","article-title":"Mathematical models for the elastic beam and their control-theoretic implications","volume":"2","author":"Russell","year":"1986","journal-title":"Semigroups. 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