No-regret Control for a degenerate population model in divergence form with missing birth rate: a nonlinear case

Abstract

We deal with a degenerate population equation in divergence form depending on time, on age and on space. In this model, the birth rate is unknown. We focus on the No-regret control and on the Low-regret control concepts of J. L. Lions treated in [Contrôle à moindres regrets des systèmes distribués, C. R. Acad. Sci.Paris Ser. I Math., SIAM J. Control Optim. ,1992, Vol 315, pp. 1253–1257] and in [Duality arguments for multi-agents least regret control, Institut de France, 1999] to treat the problem. For this purpose, we prove first the existence of the Low-regret control and the No-regret control. And we use a suitable Hilbert space to show that the No-regret control is the limit of a family of adapted Low-regret controls defined by a quadratic pertubation and previously used by Nakoulima et al. in [Perturbations à moindres regrets dans les systèmes distribués à données manquantes, C. R. Acad. Sci.Paris Ser. I Math., 2000, Vol 330, pp. 801–806]. Then we give a singular optimality system for the family of adapted Low-regret controls and for the No-regret control.