Growth conditions for the stability of a class of time-varying perturbed singular systems

Ezzine, Faten ; Hammami, Mohamed Ali

Kybernetika, Tome 58 (2022) no. 1, pp. 1-24

Abstract (VO)

In this paper, we investigate the problem of stability of linear time-varying singular systems, which are transferable into a standard canonical form. Sufficient conditions on exponential stability and practical exponential stability of solutions of linear perturbed singular systems are obtained based on generalized Gronwall inequalities and Lyapunov techniques. Moreover, we study the problem of stability and stabilization for some classes of singular systems. Finally, we present a numerical example to validate the effectiveness of the abstract results of this paper.

Détail

DOI : 10.14736/kyb-2022-1-0001

Classification : 34D20, 37B55
Keywords: linear time–varying singular systems; standard canonical form; consistent initial conditions; Gronwall inequalities; Lyapunov techniques; practical exponential stability

Bibliographie

[1] Hamed, B. Ben, Ellouze, I., Hammami, M. A.: Practical uniform stability of nonlinear differential delay equations. Mediterranean J. Math. 8 (2011), 603-616. | DOI | MR

[2] Abdallah, A. Ben, Ellouze, I., Hammami, M. A.: Practical stability of nonlinear time-varying cascade systems. J. Dynamic. Control Systems 15 (2009), 45-62. | DOI | MR

[3] Bellman, R.: The stability of solutions of linear differential equations. Duke Math. J. 10 (1943), 643-647. | DOI | MR

[4] Bainov, D., Simenov, P.: Integral Inequalities and Applications. Springer, Kluwer Academic Publishers, Congress, Dordrecht 1992. | MR

[5] Berger, T.: Bohl exponent for time-varying linear differential-algebraic equations. Int. J. Control 10 (2012), 1433-1451. | DOI | MR

[6] Berger, T., Ilchmann, A.: On the standard canonical form of time-varying linear DAEs. Quarterly Appl. Math. 71 (2013), 69-87. | DOI | MR

[7] Berger, T., Ilchmann, A.: On stability of time-varying linear differential-algebraic equations. Int. J. Control 86 (2013), 1060-1076. | DOI | MR

[8] Campbell, L.: Singular Systems of Differential Equations. Pitman Advanced Publishing Program, London 1980. | MR

[9] Campbell, S. L.: Singular Systems of Differential Equations II. Pitman Advanced Publishing Program, London 1982. | MR

[10] Caraballo, T., Ezzine, F., Hammami, M. A.: On the exponential stability of stochastic perturbed singular systems in mean square. Appl. Math. Optim. 84 (2021), 2923-2945. | DOI | MR

[11] Caraballo, T., Ezzine, F., Hammami, M., Mchiri, L.: Practical stability with respect to a part of variables of stochastic differential equations. Stochastics Int. J. Probab. Stoch. Process. 5 (2021), 647-664. | DOI | MR

[12] Caraballo, T., Ezzine, F., Hammami, M.: Partial stability analysis of stochastic differential equations with a general decay rate. J. Engrg. Math. 130 (2021), 1-17. | DOI | MR

[13] Dai, L.: Singular Control Systems. Springer-Verlag, Berlin 1989. | MR

[14] Debeljkovic, D. Lj., Jovanovic, B., Drakulic, V.: Singular system theory in chemical engineering theory: Stability in the sense of Lyapunov: A survey. Hemijska Industrija 6(2001), 260-272.

[15] Dragomir, S. S.: Some Gronwall Type Inequalities and Applications. Nova Science Publishers, Hauppauge 2003. | MR

[16] Gronwall, T. H.: Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Ann. Math. 20 (1919), 293-296. | DOI | MR

[17] Kunkel, P., Mehrmann, V.: Differential-Algebraic Equations Analysis and Numerical Solution. EMS Publishing House, Zurich 2006. | MR

[18] Luenberger, D. G.: Dynamic equation in descriptor form. IEEE Trans. Automat. Control 22 (1977), 310-319. | DOI | MR

[19] Luenberger, D. G.: Time-invariant descriptor systems. Automatica 14 (1978), 473-480. | DOI

[20] Ownes, D. H., Debeljkovic, D. Lj.: Consistency and Liapunov stability of linear descriptor systems: A geometric analysis. IMA J. Math. Control Inform. 2 (1985), 139-151. | DOI

[21] Pham, Q. C., Tabareau, N., Slotine, J. E.: A contraction theory approach to stochastic Incremental stability. IEEE Trans. Automat. Control 54 (2009), 1285-1290. | DOI | MR

[22] Rosenbrock, H. H.: Structure properties of linear dynamical systems. Int. J. Control 20 (1974), 191-202. | DOI | MR

[23] Ritt, J. F.: Systems of algebraic differential equations. Ann. Math. 36 (1935), 293-302. | DOI | MR

[24] Vrabel, R.: Local null controllability of the control-affine nonlinear systems with time-varying disturbances, Direct calculation of the null controllable region. Europ. J. Control 40 (2018), 80-86. | DOI | MR