In this paper, we analyze the stability of linear interval matrix systems in the presence of parametric uncertainties. We aim to develop mathematical conditions which allow different notions of stability within a unified framework based on structured singular values. In particular, we aim to establish both necessary and sufficient criteria for asymptotic stability, $ \mathfrak{D} $-stability, strong $ \mathfrak{D} $-stability, $ \hat{H} $-stability, and $ \mathfrak{D}(\alpha) $-stability. The proposed formulation can be viewed as an extension to classical diagonal stability with interval uncertainty, where the system parameters vary within prescribed bounds. The main results depend on vertex-based analysis combined with structured perturbation techniques and provide a way to quantify robustness margins. To illustrate the applicability of the proposed methodology, several numerical examples are presented, including cases motivated by aerospace and industrial systems. The results indicate that the proposed framework offers a noticeable reduction in computational effort while preserving the theoretical properties of existing methods.
Citation: Mutti-Ur Rehman, Ali Algefary. Asymptotic stability, $ \mathfrak{D} $-stability, strong $ \mathfrak{D} $-stability, $ \hat{H} $-stability, and $ \mathfrak{D}(\alpha) $-stability[J]. AIMS Mathematics, 2026, 11(9): 30194-30213. doi: 10.3934/math.20261196
In this paper, we analyze the stability of linear interval matrix systems in the presence of parametric uncertainties. We aim to develop mathematical conditions which allow different notions of stability within a unified framework based on structured singular values. In particular, we aim to establish both necessary and sufficient criteria for asymptotic stability, $ \mathfrak{D} $-stability, strong $ \mathfrak{D} $-stability, $ \hat{H} $-stability, and $ \mathfrak{D}(\alpha) $-stability. The proposed formulation can be viewed as an extension to classical diagonal stability with interval uncertainty, where the system parameters vary within prescribed bounds. The main results depend on vertex-based analysis combined with structured perturbation techniques and provide a way to quantify robustness margins. To illustrate the applicability of the proposed methodology, several numerical examples are presented, including cases motivated by aerospace and industrial systems. The results indicate that the proposed framework offers a noticeable reduction in computational effort while preserving the theoretical properties of existing methods.
| [1] |
E. H. Abed, Strong $D$-stability, Syst. Control Lett., 7 (1986), 207–212. https://doi.org/10.1016/0167-6911(86)90116-7 doi: 10.1016/0167-6911(86)90116-7
|
| [2] |
J. Chen, M. K. H. Fan, C. C. Yu, On $D$-stability and structured singular values, Syst. Control Lett., 24 (1995), 19–24. https://doi.org/10.1016/0167-6911(94)00036-U doi: 10.1016/0167-6911(94)00036-U
|
| [3] |
J. Chen, S.-I. Niculescu, P. Fu, Robust stability of quasi-polynomials: Frequency-sweeping conditions and vertex tests, IEEE Trans. Automat. Control, 53 (2008), 1219–1234. http://doi.org/10.1109/TAC.2008.923686 doi: 10.1109/TAC.2008.923686
|
| [4] |
J. Chen, Sufficient conditions on stability of interval matrices: Connections and results, IEEE Trans. Automat. Control, 37 (1992), 541–544. http://doi.org/10.1109/9.126595 doi: 10.1109/9.126595
|
| [5] |
D. Carlson, A new criterion for $H$-stability of complex matrices, Linear Algebra Appl., 1 (1968), 59–64. https://doi.org/10.1016/0024-3795(68)90048-7 doi: 10.1016/0024-3795(68)90048-7
|
| [6] | R. M. Chen, Y. Lan, Y. Liu, Z. Wang, Asymptotic stability of smooth solitons and multi-solitons for the Camassa-Holm equation, arXiv preprint, 2026, arXiv: 2601.17793. |
| [7] | J.-P. Casasanta, J. W. Simpson-Porco, A Lyapunov characterization of robust $D$-stability with application to decentralized integral control of LTI systems, arXiv preprint, 2026, arXiv: 2603.13608. |
| [8] |
Z. Du, Y. Kao, X. Zhao, An input delay approach to interval type-2 fuzzy exponential stabilization for nonlinear unreliable networked sampled-data control systems, IEEE Trans Syst. Man. Cyber., 51 (2021), 3488–3497. http://doi.org/10.1109/TSMC.2019.2930473 doi: 10.1109/TSMC.2019.2930473
|
| [9] |
A. C. Enthoven, K. J. Arrow, A theorem on expectations and the stability of equilibrium, Econometrica, 24 (1956), 288–293. https://doi.org/10.2307/1911633 doi: 10.2307/1911633
|
| [10] |
M. Ghorbani, M. Tavakoli-Kakhki, A. Tepljakov, E. Petlenkov, A. Farnam, G. Crevecoeur, Robust stability analysis of interval fractional-order plants with interval time delay and general form of fractional-order controllers, IEEE Control Syst. Lett., 6 (2022), 1268–1273. http://doi.org/10.1109/LCSYS.2021.3091525 doi: 10.1109/LCSYS.2021.3091525
|
| [11] |
F. Garofalo, G. Celentano, Glielmo, Stability robustness of interval matrices via Lyapunov quadratic forms, IEEE Trans. Automat. Control, 38 (1993), 281–284. http://doi.org/10.1109/9.250472 doi: 10.1109/9.250472
|
| [12] |
R. Ghosh, S. Sen, K. B. Datta, An improved method for determining the stability of interval matrices, Int. J. Syst. Sci., 31 (2000), 171–176. https://doi.org/10.1080/002077200291280 doi: 10.1080/002077200291280
|
| [13] |
D. Hershkowitz, Recent directions in matrix stability, Linear Algebra Appl., 171 (1992), 161–186. https://doi.org/10.1016/0024-3795(92)90257-B doi: 10.1016/0024-3795(92)90257-B
|
| [14] |
T. Kaczorek, Stability of interval positive fractional discrete–time linear systems, Int. J. Appl. Math. Comput. Sci., 28 (2018). http://doi.org/10.2478/amcs2018-0034 doi: 10.2478/amcs2018-0034
|
| [15] |
J. Lee, T. F. Edgar, Real structured singular value conditions for the strong $D$-stability, Syst. Control Lett., 44 (2001), 273–277. https://doi.org/10.1016/S0167-6911(01)00147-5 doi: 10.1016/S0167-6911(01)00147-5
|
| [16] |
J. G. Lu, G. Chen, Robust stability and stabilization of fractional-order interval systems: An LMI approach, IEEE Trans. Automat. Control, 54 (2009), 1294–1299. http://doi.org/10.1109/TAC.2009.2013056 doi: 10.1109/TAC.2009.2013056
|
| [17] |
J. G. Lu, Y. Q. Chen, Robust stability and stabilization of fractional-order interval systems with the fractional order $\alpha:$ The $0 < \alpha < 1$ case, IEEE Trans. Automat. Control, 55 (2010), 152–158. http://doi.org/10.1109/TAC.2009.2033738 doi: 10.1109/TAC.2009.2033738
|
| [18] | Z. Li, K. Pan, Y. Yang, Stability of interval positive systems with delays, In: 2017 IEEE 7th annual international conference on CYBER technology in automation, control, and intelligent systems, 2017,966–970. http://doi.org/10.1109/CYBER.2017.8446315 |
| [19] |
W. J. Mao, J. Chu, Quadratic stability and stabilization of dynamic interval systems, IEEE Trans. Automat. Control, 48 (2003), 1007–1012. http://doi.org/10.1109/TAC.2003.812784 doi: 10.1109/TAC.2003.812784
|
| [20] |
W. J. Mao, J. Chu, Robust $\mathfrak{D}$-stability and $D$-stabilization of dynamic interval systems, Int. J. Control Autom. Syst., 5 (2007), 594–600. http://doi.org/10.1007/s00170-006-0661-9 doi: 10.1007/s00170-006-0661-9
|
| [21] |
K. A. Moornani, M. Haeri, Robust stability testing function and Kharitonov-like theorem for fractional order interval systems, IET Control Theory Appl., 4 (2010), 2097–2108. http://doi.org/10.1049/iet-cta.2009.0485 doi: 10.1049/iet-cta.2009.0485
|
| [22] | E. Martínez-García, A matter of perspective: Mapping linear rational expectations models into finite-order VAR gorm, Globalization Institute Working Paper, 389 (2020). https://doi.org/10.24149/gwp389 |
| [23] |
E. Peral, E. Im, L. Wye, S. Lee, S. Tanelli, Y. Rahmat-Samii, et al., Radar technologies for earth remote sensing from cubesat platforms, Proc. IEEE, 106 (2018), 404–418. http://doi.org/10.1109/JPROC.2018.2793179 doi: 10.1109/JPROC.2018.2793179
|
| [24] |
O. Pastravanu, M. H. Matcovschi, Diagonal stability of interval matrices and applications, Linear Algebra Appl., 433 (2010), 1646–1658. https://doi.org/10.1016/j.laa.2010.06.016 doi: 10.1016/j.laa.2010.06.016
|
| [25] | N. Razmjooy, M. Ramezani, V. Estrela, H. J. Loschi, D. A. Nascimento, Stability analysis of the interval systems based on linear matrix inequalities, In: Proceedings of the 4th Brazilian technology symposium, Cham: Springer International Publishing, 2019, 371–378. https://doi.org/10.1007/978-3-030-16053-1_36 |
| [26] |
M. Rehman, T. H Rasulov, F. Amir, $D$-stability, strong $D$-stability and $\mu$-values, Lobachevskii J. Math., 45 (2024), 1227–1233. https://doi.org/10.1134/S1995080224600754 doi: 10.1134/S1995080224600754
|
| [27] | M. Rehman, J. Alzabut, A. Phulpoto, M. Tounsi, R. Naimi, Stability, $D$-stability, strong $D$-stability of positive linear time-invariant systems with applications, Eur. J. Pure Appl. Math., 18 (2025), 6047–6047. |
| [28] |
M. Rehman, A. Shadiyev, Interconnection between $H$-stable, $D(\alpha)$-stable, $D$-semistable matrices, and $\mu$-values, Asia Pac. J. Math., 11 (2024), 102. https://doi.org/10.28924/2291-8639-23-2025-77 doi: 10.28924/2291-8639-23-2025-77
|
| [29] |
M. Rehman, T. Farmanov, A. Abdulloyev, N. Odinayeva, H. Abdullayeva, S. S. Dexhanov, Strong $D$-stability analysis of economic models, Int. J. Anal. Appl., 23 (2025), 267. https://doi.org/10.28924/2291-8639-23-2025-267 doi: 10.28924/2291-8639-23-2025-267
|
| [30] |
T. Shen, X. Wang, Z. Yuan, Robust stability for a class of uncertain systems, Acta Automat. Sinica, 33 (2007), 426. http://doi.org/10.1360/aas-007-0426 doi: 10.1360/aas-007-0426
|
| [31] | S. Stojanovic, D. Debeljkovic, The sufficient conditions for stability of continuous and discrete large-scale time-delay interval systems, In: 2005 international conference on control and automation, 1 (2005), 347–352. http://doi.org/10.1109/ICCA.2005.1528143 |
| [32] | M. Sánchez, M. Bernal, A convex approach for reducing conservativeness of Kharitonov's-based robustness analysis, IFAC-Paperson Line, 50 (2017), 832–837. http://doi.org/10.1016/j.ifacol.2017.08.148, 20th IFAC World Congress |
| [33] |
J. Shao, X. Hou, Positive definiteness of hermitian interval matrices, Linear Algebra Appl., 432 (2010), 970–979. http://doi.org/10.1016/j.laa.2009.10.011 doi: 10.1016/j.laa.2009.10.011
|
| [34] |
N. Tan, Ö. F. Özgüven, M. M. Özyetkin, Robust stability analysis of fractional order interval polynomials, ISA T., 48 (2009), 166–172. http://doi.org/10.1016/j.isatra.2009.01.002 doi: 10.1016/j.isatra.2009.01.002
|
| [35] | Y. H. Tong, S. W. Su, Sufficient $D$-stability conditions for non-square matrices, arXiv preprint, 2024, arXiv: 2406.15440. |
| [36] |
Z. Wang, T. X. Xu, Stability and stabilizability of linear time-invariant interval systems, ISA T., 145 (2024), 273–284. https://doi.org/10.1016/j.isatra.2023.11.027 doi: 10.1016/j.isatra.2023.11.027
|
| [37] | F. Wu, Z. Shi, G. Dai, On robust stability of dynamic interval systems, Control Theory Appl., 18 (2001), 113–115. |
| [38] | C. Wang, Study on robust stability of nonlinear stochastic interval systems, In: 2018 10th International Conference on Measuring Technology and Mechatronics Automation, 2018,391–394. http://doi.org/10.1109/ICMTMA.2018.00101 |
| [39] | J. Wu, Y. Liu, Z. Wang, Asymptotic stability of solitary waves for the b-family of equations, arXiv preprint, 2025, arXiv: 2512.01225. |
| [40] |
S. Xu, J. Lam, A survey of linear matrix inequality techniques in stability analysis of delay systems, Int. J. Syst. Sci., 39 (2008), 1095–1113. http://doi.org/10.1080/00207720802300370 doi: 10.1080/00207720802300370
|
| [41] |
X. Yang, Y. Lu, B. Liu, J. Yao, Fiber ring laser temperature sensor based on liquid-filled photonic crystal fiber, IEEE Sens. J., 17 (2017), 6948–6952. http://doi.org/10.1109/JSEN.2017.2754640 doi: 10.1109/JSEN.2017.2754640
|
| [42] | R. Yedavalli, An improved extreme point solution for checking robust stability of interval matrices with much reduced vertex set and combinatorial effort, In: Proceedings of the 2001 American Control Conference. (Cat. No. 01CH37148), 5 (2001), 3902–3907. http://doi.org/10.1109/ACC.2001.946256 |
| [43] | L. K. Zhao, S. S. Fan, A new arithmetic of lower conservatism degree of robust stability for linear interval systems, In: 2008 7th world congress on intelligent control and automation, 2008, 8134–8138. http://doi.org/10.1109/WCICA.2008.4594600 |
Mutti-Ur Rehman, Ali Algefary. Asymptotic stability, $ \mathfrak{D} $-stability, strong $ \mathfrak{D} $-stability, $ \hat{H} $-stability, and $ \mathfrak{D}(\alpha) $-stability[J]. AIMS Mathematics, 2026, 11(9): 30194-30213. doi: 10.3934/math.20261196
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