Delay differential systems with time-varying delay: new directions for stability theory

Louisell, James

Kybernetika, Tome 37 (2001) no. 3, pp. 239-251

Abstract (VO)

In this paper we give an example of Markus–Yamabe instability in a constant coefficient delay differential equation with time-varying delay. For all values of the range of the delay function, the characteristic function of the associated autonomous delay equation is exponentially stable. Still, the fundamental solution of the time-varying system is unbounded. We also present a modified example having absolutely continuous delay function, easily calculating the average variation of the delay function, and then relating this average to earlier work of the author on preservation of the stability exponent in delay differential equations with time-varying delay. In this way we suggest one possible viewpoint on the conditions for Markus–Yamabe instability. Finally, we give a very brief sketch of an example of quenching of instability. To suggest a view on conditions for quenching phenomena, we relate this to earlier work of Cooke on preservation of spectral dynamics in delay systems having time-varying delay.

Détail

Classification : 34K20, 93D05, 93D99
Keywords: delay system; time-varying delay; instability

Bibliographie

[1] Chen J., Gu, G., Nett C. N.: A new method for computing delay margins for stability of linear delay systems. Systems Control Lett. 26 (1995), 107–117 | DOI | MR | Zbl

[2] Cooke K. L.: Linear functional differential equations of asymptotically autonomous type. J. Differential Equations 7 (1970), 154–174 | DOI | MR | Zbl

[3] Cooke K. L., Wiener J.: A survey of differential equations with piecewise continuous arguments. In: Delay Differential Equations and Dynamical Systems (S. Busenberg and M. Martelli, eds., Lecture Notes in Mathematics 1475), Springer–Verlag, Berlin – Heidelberg 1991, pp. 1–15 | MR | Zbl

[4] Datko R.: Lyapunov functionals for certain linear delay–differential equations in a Hilbert space. J. Math. Anal. Appl. 76 (1980), 37–57 | DOI | MR | Zbl

[5] Hale J. K., Lunel S. M. V.: Introduction to Functional Differential Equations. Springer–Verlag, New York 1993 | MR | Zbl

[6] Infante E. F., Castelan W. B.: A Lyapunov functional for a matrix difference–differential equation. J. Differential Equations 29 (1978), 439–451 | DOI | MR

[7] Kojima A., Uchida, K., Shimemura E.: Robust stabilization of uncertain time delay systems via combined internal – external approach. IEEE Trans. Automat. Control 38 (1993), 373–378 | DOI | MR | Zbl

[8] (ed.) W. Levine: The Control Handbook. CRC Press, Boca Raton 1996 | Zbl

[9] Louisell J.: Instability and quenching in delay systems having constant coefficients and time-varying delays. J. Math. Anal. Appl. Submitted

[10] Louisell J.: New examples of quenching in delay differential equations having time-varying delay. In: Proc. 5th European Control Conference, F 1023–1, Karlsruhe 1999

[11] Louisell J.: Numerics of the stability exponent and eigenvalue abscissas of a matrix delay system. In: Stability and Control of Time–Delay Systems (L. Dugard and E. I. Verriest, eds., Lecture Notes in Control and Information Sciences 228), Springer–Verlag, Berlin – Heidelberg – New York 1997, pp. 140–157 | MR

[12] Louisell J.: Growth estimates and asymptotic stability for a class of differential-delay equation having time-varying delay. J. Math. Anal. Appl. 164 (1992), 453–479 | DOI | MR | Zbl

[13] Louisell J.: A stability analysis for a class of differential-delay equation having time-varying delay. In: Delay Differential Equations and Dynamical Systems (S. Busenberg and M. Martelli, eds., Lecture Notes in Mathematics 1475), Springer–Verlag, Berlin – Heidelberg 1991, pp. 225–242 | MR

[14] Markus L., Yamabe H.: Global stability criteria for differential systems. Osaka J. Math. 12 (1960), 305–317 | MR | Zbl

[15] Marshall J. E., Gorecki H., Walton, K., Korytowski A.: Time–Delay Systems: Stability and Performance Criteria with Applications. Ellis Horwood, New York 1992 | Zbl

[16] Niculescu S.-I.: Stability and hyperbolicity of linear systems with delayed state: a matrix–pencil approach. IMA J. Math. Control Inform. 15 (1998), 331–347 | DOI | MR

[17] Niculescu S.-I., Souza C. E. de, Dugard, L., Dion J.-M.: Robust exponential stability of uncertain systems with time–varying delays. IEEE Trans. Automat. Control 43 (1998), 743–748 | DOI | MR | Zbl

[18] Repin I. M.: Quadratic Liapunov functionals for systems with delays. Prikl. Mat. Mekh. 29 (1965), 564–566 | MR

[19] Sun Y.-J., Hsieh J.-G., Yang H.-C.: On the stability of uncertain systems with multiple time–varying delays. IEEE Trans. Automat. Control 42 (1997), 101–105 | DOI | MR | Zbl

[20] Verriest E. I.: Robust stability of time varying systems with unknown bounded delays. In: Proc. 33rd IEEE Conference Decision and Control, Lake Buena Vista 1994, pp. 417–422