No-regret Control for a degenerate population model in divergence form with missing birth rate: a nonlinear case
Keywords:Population dynamics, degenerate equation, incomplete data, Low-regret control, No-regret control
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 [ContrÃ´le Ã moindres regrets des systÃ¨mes distribuÃ©s, C. R. Acad. Sci.Paris Ser. I Math., SIAM J. Control Optim. ,1992, Vol 315, pp. 1253â€“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 Ã moindres regrets dans les systÃ¨mes distribuÃ©s Ã donnÃ©es manquantes, C. R. Acad. Sci.Paris Ser. I Math., 2000, Vol 330, pp. 801â€“806]. Then we give a singular optimality system for the family of adapted Low-regret controls and for the No-regret control.
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