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

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.

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 and 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.

[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.

[3]

N. GillisM. 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.

[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.

[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.

[6] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1991. 
[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.

[8]

V. Noferini and F. Poloni, Nearest $\Omega$-stable matrix via Riemannian optimization, arXiv: 2002.07052.

[9]

F.-X. OrbandexivryY. 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.

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.

[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.

[3]

N. GillisM. 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.

[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.

[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.

[6] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1991. 
[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.

[8]

V. Noferini and F. Poloni, Nearest $\Omega$-stable matrix via Riemannian optimization, arXiv: 2002.07052.

[9]

F.-X. OrbandexivryY. 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.

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