An optimal control problem related to a 3D-chemotaxis-Navier-Stokes model

Abstract

In this paper, we study an optimal control problem associated to a 3D-chemotaxis-Navier-Stokes model. First we prove the existence of global weak solutions of the state equations with a linear reaction term on the chemical concentration equation, and an external source on the velocity equation, both acting as controls on the system. Second, we establish aregularity criterion to get global-in-time strong solutions. Finally, we prove the existence of an optimal solution, and we establish a first-order optimality condition.

Bilinear Optimal Control of the Keller–Segel Logistic Model in 2D-Domains

“…These results are based on Carleman-type estimates for the solutions of the adjoint system. Recently, a bilinear optimal control problem associated to the chemotaxis-Navier-Stokes model (without logistic source) in bounded 3D domains was examined in [18]. For the chemo-repulsion case, this problem was studied in [11,13] for 2D and 3D domains respectively, and in [12] for 2D domains with a potential nonlinear production term, by changing the production term u in the v equation of (1) by u p , with 1 < p ≤ 2.…”

Optimal control for a chemotaxis–haptotaxis model in two space dimensions

“…Remark 5.2 The first-order necessary optimality conditions for chemotaxis models have been intensively studied [10,12,23]. Recently, Colli, Signori and Sprekels [9] established both first-order necessary and second-order sufficient conditions for a tumor growth model of Cahn-Hilliard type, including chemotaxis with possibly singular potentials.…”

“…The scalar functions u = u(x, t), v = v(x, t), and w = w(x, t) represent the density of cancer cells, the concentration of enzyme, and the density of healthy tissue, respectively. Notice that in the region of where f ≥ 0 the control acts as a proliferation source of the chemical substance, and inversely, in the region of where f ≤ 0 the control acts as a degradation source of the chemical substance [23]. In this work, the function f ≥ 0 lies in a closed convex set F .…”

“…They also derived the first-order optimality conditions by using a Lagrange multipliers theorem. López-Ríos and Villamizar-Roa [23] studied an optimal control problem associated to a 3D-chemotaxis-Navier-Stokes model. Some other results can be found in [4-7, 16, 20, 22, 29, 35, 36].…”

Control Problem Related to a 2D Parabolic–Elliptic Chemo-Repulsion System with Nonlinear Production

“…Optimal control problems related with chemotaxis systems can be consulted in [12][13][14]16,[19][20][21][22][23][24]. All these works proved the existence of at least one global optimal solution and derived an optimality system, in particular obtaining first-order necessary optimality conditions.…”