Optimal Control Related to Weak Solutions of a Chemotaxis-Consumption Model

• Published in: Applied mathematics & optimization Vol. 89; no. 2; p. 48
• Main Authors: Corrêa Vianna Filho, André Luiz, Guillén-González, Francisco
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.04.2024; Springer Nature B.V
• Subjects: Boundary conditions; Calculus of Variations and Optimal Control; Optimization; Consumption; Control; Initial conditions; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Numerical and Computational Physics; Optimal control; Simulation; Systems Theory; Theoretical; Consumption; Weak solutions; 49J20; Optimal control; Energy inequality; 92C17; Chemotaxis; Primary 35K51; 35Q92
• ISSN: 0095-4616; 1432-0606
• DOI: 10.1007/s00245-024-10109-6

In the present work we investigate an optimal control problem related to the following chemotaxis-consumption model in a bounded domain Ω ⊂ R 3 : ∂ t u - Δ u = - ∇ · ( u ∇ v ) , ∂ t v - Δ v = - u s v + f v 1 Ω c , with s ≥ 1 , endowed with isolated boundary conditions and initial conditions for ( u ,  v ), being u the cell density, v the chemical concentration and f the control acting in the v -equation through the bilinear term f v 1 Ω c , in a subdomain Ω c ⊂ Ω . We address the existence of optimal control restricted to a weak solution setting, where, in particular, uniqueness of state ( u ,  v ) given a control f is not clear. Then by considering weak solutions satisfying an adequate energy inequality, we prove the existence of optimal control subject to uniformly bounded controls. Finally, we discuss the relation between the considered control problem and two other related ones, where the existence of optimal solution can not be proved.