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Some numerical results for control of 3D heat equations using Nash equilibrium
Computational and Applied Mathematics ( IF 2.998 ) Pub Date : 2021-03-22 , DOI: 10.1007/s40314-021-01488-5
Pitágoras Pinheiro de Carvalho

This article deals with some numerical strategies to solve control problems for heat equations with Dirichlet boundary conditions. We assume that we can act on the system through two controls. We use a non-cooperative bi-objective optimization strategy, to which we define the associated Nash equilibrium. More precisely, for such problems, we look for Nash equilibrium associated with optimal cost functionalities, which correspond to appropriate non-cooperative strategies. To numerically calculate the solutions, we combine finite difference methods in time and finite element methods in space. In each system presented, we seek to solve the discretized problems using three iterative methods: Fixed Point, Gradient with Fixed Step, and Conjugated Gradient, and the developed algorithms will be analyzed and compared. The data programming and computational simulations are performed in the software FreeFem ++ and Matlab.



中文翻译:

使用Nash平衡控制3D热方程的一些数值结果

本文讨论了一些数值策略,以解决带Dirichlet边界条件的热方程的控制问题。我们假设我们可以通过两个控件对系统进行操作。我们使用非合作的双目标优化策略,为此我们定义了相关的纳什均衡。更确切地说,对于此类问题,我们寻找与最佳成本功能相关的纳什均衡,这与适当的非合作策略相对应。为了对解决方案进行数值计算,我们将时间上的有限差分方法与空间上的有限元方法相结合。在提出的每个系统中,我们试图使用三种迭代方法(不动点,固定步长的梯度和共轭梯度)来解决离散问题,并将对开发的算法进行分析和比较。FreeFem ++和Matlab。

更新日期:2021-03-23
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