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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 |
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.
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. 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. |
| [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. 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. |
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. 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. |
| [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. 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. |
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