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<p>This paper deals with an optimal control problem for a chemotaxis system proposed in Luca et al. (Bull Math Biol 65(4):693–730, 2003) describing the aggregation of microglia observed in Alzheimer’s disease. In this model, the movement of cells is directed in response to two chemical signal substances, both produced by the cells, one acting as a chemoattractant, and the other as a chemorepellent. We address the problem of finding a couple of controls that provided the optimal external effects to maintain the cell-density and chemical signals close to some desired states. First, we analyze the existence and uniqueness of strong solutions for the controlled system with spatial dimension <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11590_2024_2162_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">\(N=2,\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> analyzing the effect of considering a nonlinear production of the chemoattractant substance. Subsequently, we prove the existence of global optimal solutions for the bilinear optimal control problem through the technique of minimizing sequences, and finally, using a Lagrange multipliers theorem in Banach spaces, we derive first-order necessary optimality conditions for any local optimal solution. Finally, we present a numerical experiment solving the optimal control problem by using the steep-descent gradient method, visualizing the controlled unknowns with respect to the corresponding desired states.</p>

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An optimal control problem of a 2D-attraction-repulsion chemotaxis system

  • 期刊论文

  • 发表日期：

    2025年1月4日

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  • Julio Huayta-Centeno,
  • Exequiel Mallea-Zepeda,
  • Élder J. Villamizar-Roa

摘要

This paper deals with an optimal control problem for a chemotaxis system proposed in Luca et al. (Bull Math Biol 65(4):693–730, 2003) describing the aggregation of microglia observed in Alzheimer’s disease. In this model, the movement of cells is directed in response to two chemical signal substances, both produced by the cells, one acting as a chemoattractant, and the other as a chemorepellent. We address the problem of finding a couple of controls that provided the optimal external effects to maintain the cell-density and chemical signals close to some desired states. First, we analyze the existence and uniqueness of strong solutions for the controlled system with spatial dimension \(N=2,\) N = 2 , analyzing the effect of considering a nonlinear production of the chemoattractant substance. Subsequently, we prove the existence of global optimal solutions for the bilinear optimal control problem through the technique of minimizing sequences, and finally, using a Lagrange multipliers theorem in Banach spaces, we derive first-order necessary optimality conditions for any local optimal solution. Finally, we present a numerical experiment solving the optimal control problem by using the steep-descent gradient method, visualizing the controlled unknowns with respect to the corresponding desired states.