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April  2021, 26(4): 2025-2035. doi: 10.3934/dcdsb.2020358

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

Yueh-Cheng Kuo 1, , Huan-Chang Cheng 1, , Jhih-You Syu 1, and  Shih-Feng Shieh 2,,

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  April 2021 Early access  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.

Keywords: Linear continuous-time systems, matrix approximation, stability radius, global minimizer.
Mathematics Subject Classification: 15A39, 15A60.
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, 2021, 26 (4) : 2025-2035. doi: 10.3934/dcdsb.2020358
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

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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

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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
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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

show all references

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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