Post-Doctoral Research Visit F/M Optimal control theory with applications to epidemiology

This research project is conducted in the framework of the project NOCIME (New Observation and Control Issues Motivated by Epidemiology), funded by the French National Research Agency (ANR) on the period 2024-2026 (three years). We are offering a 2-year position for a young PhD. Contracting may be immediate, and has to be achieved before January 1st, 2025.

NOCIME consortium includes researchers from Inrae (Montpellier), Inria (Paris, Lille, Metz) and IRD (Paris). The researchers in charge of the Working Package related to this position are located in Montpellier (Inrae, Campus de La Gaillarde) and Paris (Sorbonne Université). The postdoctoral supervisors are two senior researchers: Alain Rapaport (Inrae, Montpellier) and Pierre-Alexandre Bliman (Inria, Paris). The postdoctoral fellow will work mainly in Montpellier, with regular trips and contacts in Paris. Other scientific voyages (workshops, conferences) will be scheduled and funded.

The postdoctoral fellow will develop and test numerically new research results related to the topic. She/he will write scientific publications for international conferences and first-rank journals, mainly in the domains of Control theory and of Mathematical biology. He/she will present these results during international meetings and during the meetings organized for the advancement of the project NOCIME, within which he/she will be fully integrated as a collaborator.

The aim is to study various optimal control problems with unconventional criteria, and apply them to epidemiological models in continuous time, in relatively short dimension. By unconventional, we mean criteria that are not of the usual Lagrange, Mayer or Bolza form, such as crisis time, peak minimization, or maximization of the final size. The work will be both theoretical, in line with previous contributions, and numerical. In particular, we will study reformulations and/or approximations of these problems in a more classical form by extending the state vector in order to benefit from existing numerical methods (direct, Hamilton-Jacobi-Bellman, shooting methods...). For the applications, particular emphasis will be placed on the study of optimal control laws, especially in the form of state feedback. Guaranteed sub-optimality may be an alternative approach for problems where optimal state feedback is too difficult to characterize analytically. The coupling of control laws with state observers to be developed in the project could be studied in the second year of the postdoc.

References
[1] Bayen, T., Boumaza, K. and Rapaport, A. (2021) "Necessary optimality condition for the minimal time crisis relaxing transverse condition via regularization", ESAIM Control, Optimization and Calculus of Variations, Vol. 27, N. 105, online.
[2] Beard, R.W., Saridis, G.N. and Wen, J.T. (1998) "Approximate Solutions to the Time-Invariant Hamilton-Jacobi-Bellman Equation". Journal of Optimization Theory and Applications 96, pp. 589626.
[3] Bliman, P.A., Duprez, M., Privat, Y., and Vauchelet, N. (2021). Optimal immunity control and final size minimization by social distancing for the SIR epidemic model. Journal of Optimization Theory and Applications, Vol. 189, pp. 408436.
[4] Haberkorn, T. and Trélat, E. (2011) "Convergence results for smooth regularizations of hybrid non-linear optimal control problems". SIAM Journal on Control and Optimization, 49 (4), pp.1498- 1522.
[5] Lenhart, S. and Workman, J. T. (2007). "Optimal control applied to biological models". Mathematical and computational biology. Boca Raton (Fla.), London: Chapman & Hall/CRC.
[6] Molina, E. and Rapaport, A. (2022) "An optimal feedback control that minimizes the epidemic peak in the SIR model under a budget constraint", Automatica, Vol. 46, online.
[7] Sharomi, O. and Malik, T. (2017) "Optimal control in epidemiology". Annals of Operations Research 251, pp. 5571.
[8] Smirnov, A. (2008) "Necessary optimality conditions for a class of optimal control problems with discontinuous integrand", Proc. Steklov Inst. Math., vol. 262, 1, pp. 213230.
[9] Vinter R. (2005), Minimax Optimal Control. SIAM Journal on Control and Optimization, 44(3), pp. 939-968

Technical skills and level required : PhD, preferentially in Applied mathematics.

Languages : Sufficient practice of scientific English is required.

• Subsidized meals
• Partial reimbursement of public transport costs
• Leave: 7 weeks of annual leave + 10 extra days off due to RTT (statutory reduction in working hours) + possibility of exceptional leave (sick children, moving home, etc.)
• Possibility of teleworking and flexible organization of working hours
• Professional equipment available (videoconferencing, loan of computer equipment, etc.)
• Social, cultural and sports events and activities