# American Institute of Mathematical Sciences

## On the nearest stable $2\times 2$ matrix, dedicated to Prof. Sze-Bi Hsu in appreciation of his inspiring ideas

 1 Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung, 811, Taiwan 2 Department of Mathematics, National Taiwan Normal University, Taipei 116, Taiwan

* Corresponding author: Shih-Feng Shieh

Received  August 2020 Revised  October 2020 Published  December 2020

In this paper, we study the continuous-time nearest stable matrix problem: given a $2\times 2$ real matrix $A$, minimize the Frobenius norm of $A-X$, where $X$ is a stable matrix. We provide an explicit formula for the global minimizer $X_*$. The uniqueness of the minimizer is also studied.

Citation: Yueh-Cheng Kuo, Huan-Chang Cheng, Jhih-You Syu, Shih-Feng Shieh. On the nearest stable $2\times 2$ matrix, dedicated to Prof. Sze-Bi Hsu in appreciation of his inspiring ideas. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020358
##### References:
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##### References:
 [1] N. Choudhary, N. Gillis and P. Sharma, On approximating the nearest $\Omega$-stable matrix, Numer Alg. Appl., 27 (2020), e2282, 13pp. doi: 10.1002/nla.2282.  Google Scholar [2] N. Gillis, V. Mehrmann and P. Sharma, Computing the nearest stable matrix pairs, Numer. Linear Alg. Appl., 25 (2018), e2153, 16pp. doi: 10.1002/nla.2153.  Google Scholar [3] N. Gillis, M. Karow and P. Sharma, Approximating the nearest stable discrete-time system, Linear Alg. Appl., 573 (2019), 37-53.  doi: 10.1016/j.laa.2019.03.014.  Google Scholar [4] N. Gillis and P. Sharma, On computing the distance to stability for matrices using linear dissipative Hamiltonian systems, Automatica, 85 (2017), 113-121.  doi: 10.1016/j.automatica.2017.07.047.  Google Scholar [5] N. Higham, Matrix nearness problems and applications, Applications of Matrix Theory (Bradford, 1988), 1-27, Inst. Math. Appl. Conf. Ser. New Ser., 22, Oxford Univ. Press, New York, 1989.  Google Scholar [6] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1991.   Google Scholar [7] V. Mehrmann and P. Van Dooren, Optimal Robustness of Port-Hamiltonian Systems, SIAM J. Matrix Anal. Appl., 41 (2020), 134-151.  doi: 10.1137/19M1259092.  Google Scholar [8] V. Noferini and F. Poloni, Nearest $\Omega$-stable matrix via Riemannian optimization, arXiv: 2002.07052. Google Scholar [9] F.-X. Orbandexivry, Y. Nesterov and P. Van Dooren, Nearest stable system using successive convex approximations, Automatica, 49 (2013), 1195-1203.  doi: 10.1016/j.automatica.2013.01.053.  Google Scholar
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