发表机构
Università di Pavia; Weierstrass Institute for Applied Analysis and Stochastics（帕维亚大学; 魏尔斯特拉斯应用分析与随机性研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对具有双曲松弛的相场肿瘤生长模型，证明当松弛参数趋于零时，最优控制及其状态变量收敛到未松弛情形，且任意最优控制可由自适应代价泛函下的极小元逼近。
AI 中文摘要
本文研究了一类Cahn--Hilliard型相场肿瘤生长模型的最优控制问题，该模型可能具有奇异势，其中通常假设的化学势抛物型松弛被双曲型松弛所替代。我们研究了当松弛参数α趋近于零时控制问题的渐近行为。结果表明，松弛问题的相应状态变量和伴随状态变量在适当意义下收敛到未松弛情形（α=0）下的对应变量。此外，松弛问题的最优控制的弱极限可以证明是未松弛情形的最优控制。而且，对于未松弛系统的任意固定最优控制u*，可以证明它是松弛系统族{u*_α}_{α>0}的极限，其中这些松弛系统是相应最优控制问题的极小元，且原始代价泛函被所谓的“自适应”代价泛函所替代。文中还证明了u*_α满足的变分不等式（即具有自适应代价泛函的松弛问题的一阶必要最优性条件）收敛到u*对于具有原始（非自适应）代价泛函的未松弛问题所满足的相应变分不等式。
英文摘要
In this paper, we study the optimal control of a phase field tumor growth model of Cahn--Hilliard type with possibly singular potential in which the often assumed parabolic relaxation of the chemical potential is replaced by a hyperbolic one. We investigate the asymptotic behavior of the control problem as the relaxation parameter $\,α\,$ approaches zero. It is shown that the corresponding state and adjoint state variables of the relaxed problem converge in a suitable sense to their counterparts for the unrelaxed regime where $α=0$. Also, weak limits of optimal controls for the relaxed problem can be shown to be optimal controls for the unrelaxed case. Moreover, every fixed optimal control $\,u^*\,$ for the unrelaxed system turns out to be the limit of a family $\{u^*_α\}_{α>0}$ of minimizers of suitable optimal control problems for the relaxed systems, where the original cost functional is replaced by the so-called ``adapted'' cost functional. It is also shown that the variational inequalities satisfied by $u^*_α$, which constitute the first-order necessary optimality conditions of the relaxed problems with adapted cost functional, converge to the corresponding variational inequality fulfilled by $u^*$ for the unrelaxed problem with the original (nonadapted) cost functional.
CommentsKey words: tumor growth models, singular potentials, hyperbolic relaxation, optimal control, asymptotic behavior
