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: |
[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. |