The effect of delay interval on the feedback control for a turbidostat model |
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| Institution: | 1. College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China;2. Department of Mathematics, Swinburne University of Technology, Hawthorn, Victoria 3122, Australia;1. Communications Systems Department, INPT, Rabat, Morocco;2. CTTC, Barcelona, Spain;3. Future Technology Research Center, National Yunlin University of Science and Technology, Taiwan, R.O.C.;4. Department of Computer Engineering, Federal University of Ceará, Sobral, CE, Brazil;1. Maulana Abul Kalam Azad University of Technology, Nadia 741249, West Bengal, India;2. National Institute of Technology, Durgapur 713209, India;3. National Institute of Technical Teachers’ Training and Research, Kolkata 700106, India;4. Yuzuncu Yil University, Van 65080, Turkey;5. School of Computer Science & Engineering, XIM University, Bhubaneswar 751013, India;6. Global Institute of Science and Technology, Haldia 721657, West Bengal, India;1. School of Measurement and Communication, Harbin University of Science and Technology, Harbin 150080, China;2. School of Science, Harbin University of Science and Technology, Harbin 150080, China;3. School of Automation, Harbin University of Science and Technology, Harbin 150080, China;4. Heilongjiang Provincial Key Laboratory of Optimization Control and Intelligent Analysis for Complex Systems, Harbin University of Science and Technology, Harbin 150080, China;1. College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China;2. School of Mathematics, Southeast University, Nanjing 210096, China;3. Department of Mathematics, Harbin Institute of Technology at Weihai, Shandong 264209, China |
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| Abstract: | In this paper, we propose a turbidostat model with delay interval on its output using a feedback control law, aiming to investigate how the delay interval affects the feedback control of the model. The delay interval is represented by two parameters, which describe the time delay distributed in a past sub-interval. We first prove the positivity and boundedness of solutions and the permanence of the model. Then, using the input flow rate as a feedback control variable, we discuss the asymptotical stabilization of a given state (i.e., the positive equilibrium) employing the method of Lyapunov functionals. Moreover, we further study the Hopf bifurcations induced by the two delay parameters. Our theoretical and numerical results all show that the delay interval can have a significantly different effect on the dynamics of a turbidostat model from other delay types. |
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