M. Bardi and I. Capuzzo-dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Modern Birkhaser Classics, 1997.

T. Bayen and F. Mairet, Minimal time control of fed-batch bioreactor with product inhibition, Bioprocess and Biosystems Engineering, vol.21, issue.10, pp.1485-1496, 2013.
DOI : 10.1016/j.jprocont.2011.05.004
URL : https://hal.archives-ouvertes.fr/hal-00734225

B. Benyahia, A. Charfi, N. Benamar, M. H. Eran, A. Grasmick et al., HARMAND A simple model of anaerobic membrane bioreactor for control design: coupling the " AM2b " model with a simple membrane fouling dynamics, World Congress on Anaerobic Digestion: Recovering (bio) Resources for the World. AD13, 2013.

B. Bonnard and M. Chyba, Singular Trajectories and their role in Control Theory, 2002.

U. Boscain and B. Piccoli, Optimal Syntheses for Control Systems on 2-D Manifolds, 2004.

F. L. Chernous-'ko, S. A. Reshmin, and I. M. Ananievski, Control of Nonlinear Dynamical Systems: Methods and Applications, 2009.

N. G. Cogan and S. Chellamb, A method for determining the optimal back-washing frequency and duration for dead-end microfiltration, Journal of Membrane Science, vol.469, pp.410-417, 2014.
DOI : 10.1016/j.memsci.2014.06.052

N. G. Cogan, J. Li, A. R. Badireddy, and S. Chellamb, Optimal backwashing in dead-end bacterial microfiltration with irreversible attachment mediated by extracellular polymeric substances production, Journal of Membrane Science, vol.520, pp.337-344, 2016.
DOI : 10.1016/j.memsci.2016.08.001

T. Faulwasser and D. Bonvin, Exact turnpike properties and economic NMPC, European Journal of Control, vol.35, pp.34-41, 2017.
DOI : 10.1016/j.ejcon.2017.02.001

T. Faulwasser, M. Korda, C. Jones, and D. Bonvin, On turnpike and dissipativity properties of continuous-time optimal control problems, Automatica, vol.81, pp.297-304, 2017.
DOI : 10.1016/j.automatica.2017.03.012

G. Ferrero, H. Moncl´usmoncl´-moncl´us, G. Buttiglieri, J. Comas, and I. Rodriguez-roda, Automatic control system for energy optimization in membrane bioreactors, Desalination, vol.268, issue.1-3, pp.276-280, 2011.
DOI : 10.1016/j.desal.2010.10.024

G. Ferrero, I. Rodriguez-roda, and J. Comas, Automatic control systems for submerged membrane bioreactors: A state-of-the-art review, Water Research, vol.46, issue.11, pp.3421-3433, 2011.
DOI : 10.1016/j.watres.2012.03.055

A. F. Filippov, Differential Equatiions with Discontinuous Righthand Sides, 1988.
DOI : 10.1007/978-94-015-7793-9

H. A. Frankowska and . Ochal, On singularities of value function for Bolza optimal control problem, Journal of Mathematical Analysis and Applications, vol.306, issue.2, pp.714-729, 2005.
DOI : 10.1016/j.jmaa.2004.10.003

N. Kalboussi, J. Harmand, N. Ben-amar, and F. Ellouze, A comparative study of three membrane fouling models -Towards a generic model for optimization purposes, CARI'2016, pp.10-16, 2016.

N. Kalboussi, A. Rapaport, T. Bayen, N. Ben-amar, F. El-louze et al., Optimal control of a membrane filtration system, 20th World Congress of the International Federation of Automatic Control, 2017.
DOI : 10.1016/j.ifacol.2017.08.1554
URL : https://hal.archives-ouvertes.fr/hal-01562522

N. Kalboussi, J. Harmand, A. Rapaport, T. Bayen, N. Ben-amar et al., Optimal control of physical backwash strategy - towards the enhancement of membrane filtration process performance, Journal of Membrane Science, vol.545, pp.38-48, 2018.
DOI : 10.1016/j.memsci.2017.09.053
URL : https://hal.archives-ouvertes.fr/hal-01591027

S. N. Hong, H. W. Zhao, and R. W. Dimassimo, A new approach to backwash initiation in membrane systems, U.S. Patent No, pp.7-459

. Washington, DC: U.S. Patent and Trademark Office, 2008.

E. B. Lee and L. Markus, Foundations of Optimal Control Theory, SIAM series in applied mathematics, 1967.

L. Pontryagin, V. Boltyanski, R. Gamkrelidze, and E. Michtchenko, The Mathematical Theory of Optimal Processes, The Macmillan Company, 1964.

A. Rapaport and P. Cartigny, Turnpike theorems by a value function approach, ESAIM Control, Optimizationa and Calculus of variations, pp.123-141, 2004.

A. Robles, M. V. Ruano, J. Ribes, and J. Ferrero, Advanced control system for optimal filtration in submerged anaerobic MBRs (SAnMBRs), Journal of Membrane Science, vol.430, pp.330-341, 2013.
DOI : 10.1016/j.memsci.2012.11.078

P. J. Smith, S. Vigneswaran, H. H. Ngo, R. Ben-aim, and H. Nguyen, A new approach to backwash initiation in membrane systems, Journal of Membrane Science, vol.278, issue.1-2, pp.381-389, 2006.
DOI : 10.1016/j.memsci.2005.11.024

E. Trélattr´trélat and E. Zuazua, The turnpike property in finite-dimensional nonlinear optimal control, Journal of Differential Equations, vol.258, issue.1, pp.81-114, 2015.
DOI : 10.1016/j.jde.2014.09.005

A. Vargas, I. Moreno-andrade, G. And, and . Buitr´on, Controlled backwashing in a membrane sequencing batch reactor used for toxic wastewater treatment, Journal of Membrane Science, vol.320, issue.1-2, pp.185-190, 2008.
DOI : 10.1016/j.memsci.2008.03.073

J. Yong and X. Y. Zhou, Stochastic controls, Hamiltonian Systems and HJB equations, 1999.

M. Zelikin and V. Borisov, Theory of Chattering Control with applications to Astronautics, Robotics, Economics, and Engineering, Systems & Control: Foundations & Applications, Birkhuser, 1994.