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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