Abstract
Considering a nonlinear dynamical system, we study the nonlinear infinite-dimensional system obtained by grafting an operatorAand an age structure. This system is such that the nonlinearity is at the level of births. We show that there is a timeTdependent on the constraints on the age and the observability minimal timeT0of the pairA,B(Bis the control operator), from which the system is null controllable. We first establish an observability inequality useful for the proof of the null controllability of an auxiliary system. We also apply Schauder’s fixed point in the proof of the null controllability of the nonlinear system..
Highlights
Introduction and Main ResultsIn this paper, we study the null controllability of an infinitedimensional nonlinear system describing the dynamics of age-structured population.Let V, H, and U be a four separable Hilbert space
We study the null controllability of an infinitedimensional nonlinear system describing the dynamics of age-structured population
Considering a nonlinear infinite dimensional system obtained by grafting an operator A and an age structure, we show there is a time T dependent on the constraints on the age and the observability minimal time T′ of the pair (A, B) (B is the control operator) from which the system is null controllable
Summary
We study the null controllability of an infinitedimensional nonlinear system describing the dynamics of age-structured population. In [3], Traoreproved the nonlinear Lotka–McKendrick is null controllable except for a small interval of ages near zero where the controls is localized with respect to the space variable but active for all ages. One of the advantages of this method is that the conditions A0 μ0(a)da +∞ on the mortality function do not interfere in establishing of the observability inequality which allows us to have a global controllability, which is not the case in [3, 9]. The control is localized with respect to the age variable, and we obtain the state of the system null for all ages.
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