| Issue |
ESAIM: COCV
Volume 25, 2019
|
|
|---|---|---|
| Article Number | 7 | |
| Number of page(s) | 22 | |
| DOI | https://doi.org/10.1051/cocv/2017075 | |
| Published online | 11 April 2019 | |
The bang-bang property of time and norm optimal control problems for parabolic equations with time-varying fractional Laplacian*
1
Laboratory of Information & Control Technology, Ningbo Institute of Technology, Zhejiang University,
Ningbo
315100, P.R. China.
2
Department of Mathematics, Wuhan University of Technology,
Wuhan
430070, P.R. China.
** Corresponding author: zhangl@whut.edu.cn
Received:
16
April
2017
Accepted:
20
November
2017
In this paper, we establish the bang-bang property of time and norm optimal control problems for parabolic equations governed by time-varying fractional Laplacian, evolved in a bounded domain of ℝd. We firstly get a quantitative unique continuation at one point in time for parabolic equations governed by time-varying fractional Laplacian. Then, we establish an observability inequality from measurable sets in time for solutions of the above-mentioned equations. Finally, with the aid of the observability inequality, the bang-bang property of time and norm optimal control problems can be obtained.
Key words: Time optimal control / norm optimal control / bang-bang property / observability estimates / measurable sets / fractional Laplacian
This work was partially supported by the National Natural Science Foundation of China (61374096, 61573012), the Natural Science Foundation of Hubei Province (2014CFB337), the Natural Science Foundation of Zhejiang Province (LY17C190008), the Foundation of China Scholarship Council and the Ningbo Natural Science Foundation (2014A610185).
© EDP Sciences, SMAI 2019
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