New criterion of stability for time-varying dynamical systems: application to spring-mass-damper model. (English) Zbl 07656148
Summary: In this paper, we investigate the problem of stability with respect to a part of variables of nonlinear time-varying systems. We derive some sufficient conditions that guarantee exponential stability and practical exponential stability with respect to a part of the variables of perturbed systems based on Lyapunov techniques where converse theorems are stated. Furthermore, illustrative examples to show the usefulness and applicability of the theory of stability with respect to a part of variables are provided. In particular, we show that our approach can be applied to the spring-mass-damper model.
MSC:
| 93E03 | Stochastic systems in control theory (general) |
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
Keywords:
nonlinear-time varying systems; perturbed systems; stability with respect to a part of variables; nontrivial solution; practical stability with respect to a part of variables; Lyapunov techniquesReferences:
| [1] | Ahmad, A.; Adly, S.; Terraz, O.; Ghazanfarpour, D., EG UK Theory and Practice of Computer Graphics, Spring Systems (2017), Ik Soo Lim and David Duce |
| [2] | Awad, E.; Culick, F. E.C., On the existence and stability of limit cycles for longitunical acoustic modes in a combustion chamber, Combust. Sci. Technol., 195 (1986) |
| [3] | Bernstein, D. S., From infancy to potency: Lyapunov’s second method and the past, present, and future of control theory, IEEE Conf, Decision and Control (2001), Orlando, FL |
| [4] | Bupp, R. T.; Bernstein, D. S., A benchmark problem for nonlinear and robust control, Int. J. Robust and Nonlinear Control., 8, 307 (1998) |
| [5] | Ben Hamed, B.; Ellouze, I.; Hammami, M. A., Practical uniform stability of nonlinear differential delay equations, Mediterr. J. Math., 8, 603 (2011) · Zbl 1238.34137 |
| [6] | Caraballo, T.; Hammami, M. A.; Mchiri, L., Practical exponential stability of impulsive stochastic functional differential equations, IEEE Control Syst. Lett., 109, 43 (2017) · Zbl 1377.93167 |
| [7] | Caraballo, T.; Ezzine, F.; Hammami, M. A.; Mchiri, L., Practical stability with respect to a part of variables of stochastic differential equations, Stochastics An International Journal of Probability and Stochastic Processes, 6, 1 (2020) |
| [8] | Caraballo, T.; Ezzine, F.; Hammami, M. A., Partial stability analysis of stochastic differential equations with a general decay rate, Journal of Engineering Mathematics, 130, 4 (2021) · Zbl 1478.93706 |
| [9] | Caraballo, T.; Ezzine, F.; Hammami, M. A., New stability criteria for stochastic perturbed singular systems in mean square, Nonlinear Dyn., 105, 241 (2021) · Zbl 1537.93533 |
| [10] | Caraballo, T.; Ezzine, F.; Hammami, M. A., On the exponential stability of stochastic perturbed singular systems in mean square, Appl. Math. Optim., 84, 1 (2021) · Zbl 1472.93152 |
| [11] | Caraballo, T.; Ezzine, F.; Hammami, M. A., Practical stability with respect to a part of the variables of stochastic differential equations driven by G-Brownian motion, J. Dyn. Control Syst., 33, 1 (2021) |
| [12] | Corless, M., Guaranteed rates of exponential convergence for uncertain systems, J. Optim. Theory Appl., 64, 481 (1990) · Zbl 0682.93040 |
| [13] | Ezzine, F.; Hammami, M. A., Growth conditions for the stability of a class of time—varying perturbed singular systems, Kybernetika, 58, 1 (2022) · Zbl 1524.34032 |
| [14] | Haddad, W. M.; Chellaboina, V., Nonlinear Dynamical Systems and Control (2007), Princeton University Press |
| [15] | Karafyllis, I.; Jiang, Z. P., Stability and Stabilization of Nonlinear Systems (2011), Springer: Springer London · Zbl 1243.93004 |
| [16] | Khalil, H. K., Nonlinear Systems (1996), Mac-Millan |
| [17] | Lum, K. Y.; Bernstein, D. S.; Coppola, V. T., Global stabilization of the spinning top with mass imbalance, Dyn. Stab. Syst., 10, 339 (1995) · Zbl 0837.93048 |
| [18] | Naifar, O.; Ben Makhlouf, A.; Hammami, M. A.; Ouali, A., State feedback control law for a class of nonlinear time-varying system under unknown time-varying delay, Nonlinear Dyn., 82, 349 (2015) · Zbl 1348.93127 |
| [19] | Pham, Q. C.; Tabareau, N.; Slotine, J. E., A contraction theory approach to stochastic Incremental stability, IEEE Trans. Automat. Contr., 54, 1285 (2009) · Zbl 1367.60073 |
| [20] | Rouche, N.; Habets, P.; Lalog, M., Stability Theory by Liapunov’s Direct Method (1977), Springer: Springer New York · Zbl 0364.34022 |
| [21] | Rymanstev, V. V., On the stability of motions with respect to part of variables Vestnik Moscow University, Ser. Math. Mech., 4, 9 (1957) |
| [22] | Sinitsyn, V. A., On stability of solution in inertial navigation problem, (Certain Problems on Dynamics of Mechanical Systems (1991), Moscow), 46 |
| [23] | Sontag, E. D., Input to state stability: Basic concepts and results, Nonlinear and Optimal Control Theory: Lectures given at the C.I.M.E. (2008) · Zbl 1175.93001 |
| [24] | Sontag, E.; Wang, Y., On characterizations of input-to-state stability with respect to compact sets, IFAC Nonlinear Control Systems Design (1995), Tahoe City, California, USA |
| [25] | Vorotnikov, V. I., Partial Stability and Control (1998), Birkhäuser: Birkhäuser Boston · Zbl 0891.93004 |
| [26] | Zubov, V. I., The Dynamics of Controlled Systems (1982), Moscow, Vysshaya Shkola · Zbl 0535.93001 |
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