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.
Principales activités
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