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Optimal control related to weak solutions of a chemotaxis-consumption model

Francisco Guillén-González, André Luiz Corrêa Vianna Filho

Abstract

In the present work we investigate an optimal control problem related to the following chemotaxis-consumption model in a bounded domain $Ω\subset \mathbb{R}^3$: $$\partial_t u - Δu = - \nabla \cdot (u \nabla v), \quad \partial_t v - Δv = - u^s v + f \,v\, 1_{Ω_c},$$ with $s \geq 1$, 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 $Ω_c \subset Ω$. 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.

Optimal control related to weak solutions of a chemotaxis-consumption model

Abstract

In the present work we investigate an optimal control problem related to the following chemotaxis-consumption model in a bounded domain : with , endowed with isolated boundary conditions and initial conditions for , being the cell density, the chemical concentration and the control acting in the -equation through the bilinear term , in a subdomain . We address the existence of optimal control restricted to a weak solution setting, where, in particular, uniqueness of state given a control 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.
Paper Structure (14 sections, 19 theorems, 185 equations)

This paper contains 14 sections, 19 theorems, 185 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminary results
  4. Existence of the controlled problem
  5. A Lînfty function bounding v_m from above
  6. Existence for the controlled truncated problem and the first uniform estimates
  7. Energy inequality
  8. m-independent estimates and passage to the limit as m goes to infinity
  9. m-independent estimates for grad(v_m)
  10. Case s in [1,2)
  11. Case s geq 2
  12. Existence of an Optimal Control
  13. Proof of Theorem \ref{['teo_existencia_controle_otimo']}
  14. Proof of Theorem \ref{['teo_relacao_entre_problemas_de_minimizacao']}

Key Result

Theorem 1.3

Given $f \in L^q(Q)$ ($q > 5/2$), there is a non-negative weak solution $(u,v)$ of problema_P_controlado satisfying the following energy inequality for $a.e.$$t_1,t_2 \in [0,T]$. Here, $\mathcal{K}(\| {f} \|_{L^q(Q)},\| {v_0} \|_{W^{2-2/q,q}(\Omega)})$ is a continuous and increasing function with respect to $\| {f} \|_{L^q(Q)}$ and $\beta > 0$ is a constant, independent of $(u,v,f)$. Moreover,

Theorems & Definitions (20)

  • Definition 1.1: Weak Solution of \ref{['problema_P_controlado']}
  • Theorem 1.3: Existence of energy inequality weak solutions
  • Theorem 1.5: Existence of optimal control
  • Theorem 1.7
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5: ViannaGuillen2023uniform
  • Lemma 2.6
  • ...and 10 more