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Numerical Stackelberg--Nash Control for the Heat Equation
SIAM Journal on Scientific Computing ( IF 3.1 ) Pub Date : 2020-09-09 , DOI: 10.1137/19m1253320
Pitágoras P. de Carvalho , Enrique Fernández-Cara

SIAM Journal on Scientific Computing, Volume 42, Issue 5, Page A2678-A2700, January 2020.
This paper deals with a strategy to solve numerically control problems of the Stackelberg--Nash kind for heat equations with Dirichlet boundary conditions. We assume that we can act on the system through several controls, respecting an order and a hierarchy: a first control (the leader) is assumed to choose the policy; then, a Nash equilibrium pair, determined by the choice of the leader and corresponding to a noncooperative multiple-objective optimization strategy, is found (these are the followers). Our method relies on a formulation inspired by the work of Fursikov and Imanuvilov. More precisely, we minimize over the class of admissible null controls a functional that involves weighted integrals of the state and the control, with weights that blow up at the final time. The use of the weights is crucial to ensure the existence of the controls and the associated state in a reasonable space. We present several mixed formulations of the problems and, then, associated mixed finite element approximations that are easy to handle. In a final step, we exhibit some numerical experiments making use of the Freefem++ package.


中文翻译:

热方程的数值Stackelberg-Nash控制

SIAM科学计算杂志,第42卷,第5期,第A2678-A2700页,2020年1月。
本文提出了一种解决具有Dirichlet边界条件的热方程组Stackelberg-Nash数值控制的策略。我们假设我们可以通过几个控件来操作系统,同时遵守顺序和层次结构:假定第一个控件(领导者)选择策略;然后,找到一个Nash均衡对,该均衡对由领导者的选择决定,并且对应于非合作式多目标优化策略(以下为跟随者)。我们的方法依赖于Fursikov和Imanuvilov的工作启发的公式化。更准确地说,我们在可允许的null控件类别中最小化了一种功能,该功能涉及状态和控件的加权积分,并且权重在最终时间爆炸。权重的使用对于确保控件和相关状态在合理空间中的存在至关重要。我们提出了问题的几种混合形式,然后给出了易于处理的相关混合有限元逼近。最后,我们展示了一些使用Freefem ++软件包的数值实验。
更新日期:2020-10-16
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