American Institute of Mathematical Sciences

Advanced
  • Previous Article
    A model for transmission of partial resistance to anti-malarial drugs
  • MBE Home
  • This Issue
  • Next Article
    Theoretical modeling of RF ablation with internally cooled electrodes: Comparative study of different thermal boundary conditions at the electrode-tissue interface
2009, 6(3): 629-647. doi: 10.3934/mbe.2009.6.629

Feedback stabilization for a chemostat with delayed output

Gonzalo Robledo 1,

1. 

Faculty of Mechanical Engineering, Technion Israel Institute of Technology, Haifa, 32000, Israel

Received  October 2008 Revised  January 2009 Published  June 2009

We apply basic tools of control theory to a chemostat model that describes the growth of one species of microorganisms that consume a limiting substrate. Under the assumption that available measurements of the model have fixed delay $\tau>0$, we design a family of feedback control laws with the objective of stabilizing the limiting substrate concentration in a fixed level. Effectiveness of this control problem is equivalent to global attractivity of a family of differential delay equations. We obtain sufficient conditions (upper bound for delay $\tau>0$ and properties of the feedback control) ensuring global attractivity and local stability. Illustrative examples are included.
Keywords: control theory, chemostat., differential delay equations, global stability.
Mathematics Subject Classification: Primary: 93D15, 92D25; Secondary: 34K2.
Citation: Gonzalo Robledo. Feedback stabilization for a chemostat with delayed output. Mathematical Biosciences & Engineering, 2009, 6 (3) : 629-647. doi: 10.3934/mbe.2009.6.629
[1]

Anatoli F. Ivanov, Musa A. Mammadov. Global asymptotic stability in a class of nonlinear differential delay equations. Conference Publications, 2011, 2011 (Special) : 727-736. doi: 10.3934/proc.2011.2011.727
[2]

Leonid Berezansky, Elena Braverman. Stability of linear differential equations with a distributed delay. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1361-1375. doi: 10.3934/cpaa.2011.10.1361
[3]

Eduardo Liz, Gergely Röst. On the global attractor of delay differential equations with unimodal feedback. Discrete & Continuous Dynamical Systems, 2009, 24 (4) : 1215-1224. doi: 10.3934/dcds.2009.24.1215
[4]

Jan Čermák, Jana Hrabalová. Delay-dependent stability criteria for neutral delay differential and difference equations. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4577-4588. doi: 10.3934/dcds.2014.34.4577
[5]

Fathalla A. Rihan, Yang Kuang, Gennady Bocharov. From the guest editors: "Delay Differential Equations: Theory, Applications and New Trends". Discrete & Continuous Dynamical Systems - S, 2020, 13 (9) : i-iv. doi: 10.3934/dcdss.2020404
[6]

Bao-Zhu Guo, Li-Ming Cai. A note for the global stability of a delay differential equation of hepatitis B virus infection. Mathematical Biosciences & Engineering, 2011, 8 (3) : 689-694. doi: 10.3934/mbe.2011.8.689
[7]

Qi Lü, Xu Zhang. A concise introduction to control theory for stochastic partial differential equations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021020
[8]

Zhenyu Lu, Junhao Hu, Xuerong Mao. Stabilisation by delay feedback control for highly nonlinear hybrid stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4099-4116. doi: 10.3934/dcdsb.2019052
[9]

Wei Mao, Yanan Jiang, Liangjian Hu, Xuerong Mao. Stabilization by intermittent control for hybrid stochastic differential delay equations. Discrete & Continuous Dynamical Systems - B, 2022, 27 (1) : 569-581. doi: 10.3934/dcdsb.2021055
[10]

Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete & Continuous Dynamical Systems, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295
[11]

Samuel Bernard, Fabien Crauste. Optimal linear stability condition for scalar differential equations with distributed delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1855-1876. doi: 10.3934/dcdsb.2015.20.1855
[12]

Eugen Stumpf. Local stability analysis of differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 3445-3461. doi: 10.3934/dcds.2016.36.3445
[13]

Cemil Tunç. Stability, boundedness and uniform boundedness of solutions of nonlinear delay differential equations. Conference Publications, 2011, 2011 (Special) : 1395-1403. doi: 10.3934/proc.2011.2011.1395
[14]

Samuel Bernard, Jacques Bélair, Michael C Mackey. Sufficient conditions for stability of linear differential equations with distributed delay. Discrete & Continuous Dynamical Systems - B, 2001, 1 (2) : 233-256. doi: 10.3934/dcdsb.2001.1.233
[15]

Gang Huang, Yasuhiro Takeuchi, Rinko Miyazaki. Stability conditions for a class of delay differential equations in single species population dynamics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2451-2464. doi: 10.3934/dcdsb.2012.17.2451
[16]

Stéphane Junca, Bruno Lombard. Stability of neutral delay differential equations modeling wave propagation in cracked media. Conference Publications, 2015, 2015 (special) : 678-685. doi: 10.3934/proc.2015.0678
[17]

Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3167-3197. doi: 10.3934/dcdsb.2017169
[18]

Songbai Guo, Wanbiao Ma. Global behavior of delay differential equations model of HIV infection with apoptosis. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 103-119. doi: 10.3934/dcdsb.2016.21.103
[19]

Sebastián Buedo-Fernández. Global attraction in a system of delay differential equations via compact and convex sets. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 3171-3181. doi: 10.3934/dcdsb.2020056
[20]

Mazyar Ghani Varzaneh, Sebastian Riedel. A dynamical theory for singular stochastic delay differential equations Ⅱ: nonlinear equations and invariant manifolds. Discrete & Continuous Dynamical Systems - B, 2021, 26 (8) : 4587-4612. doi: 10.3934/dcdsb.2020304

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (17)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy