American Institute of Mathematical Sciences

September  2018, 8(3&4): 1081-1095. doi: 10.3934/mcrf.2018046

Optimal actuator location of the minimum norm controls for stochastic heat equations

 1 School of Mathematics and Statistics, School of Information Science and Engineering, Central South University, Changsha 410075, China 2 Department of Mathematics, California State University, Los Angeles, Los Angeles, CA 90032, USA

* Corresponding author: Jie Zhong (jiezhongmath@gmail.com)

Dedicated to Professor Jiongmin Yong on the Occasion of His 60th Birthday

Received  September 2017 Revised  February 2018 Published  September 2018

Fund Project: The first author is supported in part by the National Natural Science Foundation of China, China Postdoctoral Science Foundation and Central South University Postdoctoral Science Foundation.

In this paper, we study the approximate null controllability for the stochastic heat equation with the control acting on a measurable subset, and the optimal actuator location of the minimum norm controls. We formulate a relaxed optimization problem for both actuator location and its corresponding minimum norm control into a two-person zero sum game problem and develop a sufficient and necessary condition for the optimal solution via Nash equilibrium. At last, we prove that the relaxed optimal solution is an optimal actuator location for the classical problem.

Citation: Donghui Yang, Jie Zhong. Optimal actuator location of the minimum norm controls for stochastic heat equations. Mathematical Control & Related Fields, 2018, 8 (3&4) : 1081-1095. doi: 10.3934/mcrf.2018046
References:

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