Null controllability of a coupled model in population dynamics

Null controllability of a retarded population dynamics model with interior degeneracy

“…Proof. Before we begin this proof, we highlight that the inequality (2.27) related to system (2.26) is a consequence of the same inequality realized for the intermediate adjoint model (2.25) and the assumptions (1.7) verified by the mortality rate 𝜇 (see the end of the proof of [17,Proposition 3.1]). Now, consider ∀(t, a, 𝜃, x) ∈ Q −r , the function 𝜈 is stated by 𝜈(t, a, 𝜃, x) ∶= e s𝜑(t,a,𝜃,x) u(t, a, x), (2.28) where u, in a first moment, is a solution of system (2.25).…”

“…To prove Lemma 2.7, one can employ a similar tool used for [17,Lemma 3.2]. Now, to have an accuracy on our calculus and following the expression of 𝜏 r stated in (2.22) as well as the ones of the weight functions Θ r and 𝜓, we will distinguish between the cases when t ∈ (0, 2r) and t ∈ (2r, T).…”

“…The population dynamics models without delay in their different aspects attracted many authors and were investigated from many sides (see, e.g., [1][2][3][4][5][6][7][8][9]). Among those questions, we find the null controllability or in general the controllability problems for age and space structured population dynamics models, which where studied intensively in the literature, namely, [10][11][12][13][14][15] for the non degenerate case as well as for the degenerate case treated for example in [16,17]. Such studies are based on the investigations done respectively for null controllability of heat equation both in the non-degenerate case assessed for instance in [18] and the degenerate parabolic operator appraised in [19][20][21][22][23][24][25][26].…”

“…Such studies are based on the investigations done respectively for null controllability of heat equation both in the non-degenerate case assessed for instance in [18] and the degenerate parabolic operator appraised in [19][20][21][22][23][24][25][26]. For further details, we advise the reader to take a glance in introduction of [27].…”

“…The result (1.4) is gotten under the conditions that all natural rates possess an L ∞ -regularity, and this will avoid us the use of the fixed point technique needed in [15,17]. Another striking difference with the cited references is that our model is an age and space structured and likewise the degeneracy occurring on the interior of the gene type domain (0, 1) for the dispersion coefficient k, that is, ∃x 0 ∈ (0, 1) such that k(x 0 ) = 0; for example, k(x) = |x − x 0 | 𝛼 , where 𝛼 can be taken in [0, 2).…”