论文标题
用于整数最佳控制的输入正则化，并应用用于控制毛弹性和孔隙弹性系统的应用
Input Regularization for Integer Optimal Control in BV with Applications to Control of Poroelastic and Poroviscoelastic Systems
论文作者
论文摘要
我们重新审视一类整数最佳控制问题，在ARXIV中提出并分析了信任区域方法：2106.13453V3 [Math.oc]。尽管ARXIV中提出的算法：2106.13453V3 [MATH.OC]成功地解决了所考虑的优化问题类别，但其收敛分析需要限制性规律性假设。有许多涉及部分微分方程的整数最佳控制问题的示例，其中不满足这些规律性假设。在本文中，我们提供了一种绕过限制性规则假设的方法，通过通过缓慢引入对控制输入的额外正规化并证明$γ$ -Convergence-type在驱动器的支持参数驱动到零时。在流体流过生物力学中出现的可变形多孔培养基方程的情况下，我们强调了该理论的适用性。我们表明，在Poro-Visco-弹性系统的情况下，规律性假设是违反的，因此需要使用本文中介绍的控制输入的正则化。相关的数值结果表明，虽然同质性可以帮助找到更好的目标值和较低的省点的点，但没有输入正则化的算法的实际性能可能与同型相当。
We revisit a class of integer optimal control problems for which a trust-region method has been proposed and analyzed in arXiv:2106.13453v3 [math.OC]. While the algorithm proposed in arXiv:2106.13453v3 [math.OC] successfully solves the class of optimization problems under consideration, its convergence analysis requires restrictive regularity assumptions. There are many examples of integer optimal control problems involving partial differential equations where these regularity assumptions are not satisfied. In this article we provide a way to bypass the restrictive regularity assumptions by introducing an additional partial regularization of the control inputs by means of mollification and proving a $Γ$-convergence-type result when the support parameter of the mollification is driven to zero. We highlight the applicability of this theory in the case of fluid flows through deformable porous media equations that arise in biomechanics. We show that the regularity assumptions are violated in the case of poro-visco-elastic systems, and thus one needs to use the regularization of the control input introduced in this article. Associated numerical results show that while the homotopy can help to find better objective values and points of lower instationarity, the practical performance of the algorithm without the input regularization may be on par with the homotopy.
