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Pure Mathematics 2025
拟线性斑秃趋化系统的最优控制问题
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Abstract:
在本文中，我们研究了三组式趋化系统的齐次Neumann初始边界值问题，该系统描述了斑秃的时空动态。与常规的趋化模型相比，该系统的一个显著特征是CD8+ T细胞在CD4+ T细胞的帮助下以非线性方式增殖。我们通过Leray-Schauder不动点定理证明了该系统强解的存在性、唯一性和正则性，随后建立了最优控制系统，推导出了系统全局最优解的存在性，并在Banach空间中利用拉格朗日乘子定理研究最优控制问题的一阶必要最优性条件，最后，我们得到了拉格朗日乘子的正则性结果。
In this paper, we investigate the problem of homogeneous Neumann initial boundary values for a three-group chemotaxis system that describes the spatiotemporal dynamics of alopecia areata. To compare with previous chemotaxis models, a distinctive feature of this system is that CD8+ T cells additionally proliferate in a nonlinear manner with the help of CD4+ T cells. We prove the existence, uniqueness and regularity of the strong solution of the system by the Leray-Schauder fixed point theorem, then establish the optimal control system, deduce the existence of the global optimal solution of the system, and use the Lagrange multiplier theorem in Banach space to study the first-order necessary optimality condition of the optimal control problem, and finally, we obtain the regularity result of the Lagrange multiplier.
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