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Fractional order PID control is typically studied in the frequency domain, but for an uncertain system with time-varying delay, it is more convenient to conduct the study in the time domain. A novel Lyapunov\u2013Krasovskii function is constructed. Based on this and a fractional order integral matrix inequality, several corresponding less conservative sufficient conditions are derived to guarantee the FTB of the system. By employing LQR and LMI approaches, the proposed conditions can be solved using some standard numerical packages. Through fine-tuning the conditions, the derived results can also be effectively applied to asymptotic stability analysis. Two numerical examples are presented, which further validate the introduced theoretical formulation and highlight its effectiveness through result comparisons.<\/jats:p>","DOI":"10.1177\/09596518251318904","type":"journal-article","created":{"date-parts":[[2025,2,19]],"date-time":"2025-02-19T06:58:42Z","timestamp":1739948322000},"page":"1034-1047","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":0,"title":["A cost guaranteed fractional PID control strategy for fractional uncertain time-varying delay systems on finite-time boundedness analysis"],"prefix":"10.1177","volume":"239","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2504-7780","authenticated-orcid":false,"given":"Yitong","family":"Jin","sequence":"first","affiliation":[{"name":"Architectural Engineering Institute, Jiangsu College of Engineering and Technology, Nantong, P.R. 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