### On the Stability Analysis of Delay Differential-Algebraic Equations

## Main Article Content

## Abstract

The stability analysis of linear time invariant delay differential- algebraic equations (DDAEs) is analyzed. Examples are delivered to demonstrate that the eigenvalue-based approach to analyze the exponential stability of dynamical systems is not valid for an arbitrarily high index system, and hence, a new concept of weak exponential stability (w.e.s) is proposed. Then, we characterize the w.e.s in term of a spectral condition for some special classes of DDAEs.

*Keywords:* Differential-algebraic equation, time delay, exponential stability, weak stability, simultaneous triangularizable.

References

[1] S. L. Campbell. Nonregular 2D descriptor delay systems. IMA J. Math. Control Appl., 12:57–67, 1995.

[2] Nguyen Huu Du, Vu Hoang Linh, Volker Mehrmann, and Do Duc Thuan. Stability and robust stability of linear time-invariant delay differential- algebraic equations. SIAM J. Matr. Anal. Appl., 34(4):1631–1654, 2013.

[3] Phi Ha and Volker Mehrmann. Analysis and reformulation of linear delay differential-algebraic equations. Electr. J. Lin. Alg., 23:703–730, 2012.

[4] Phi Ha and Volker Mehrmann. Analysis and numerical solution of linear delay differential-algebraic equations. BIT, 56:633 – 657, 2016.

[5] S. L. Campbell and V. H. Linh. Stability criteria for differential-algebraic equations with multiple delays and their numerical solutions. Appl. Math Comput., 208(2):397 – 415, 2009.

[6] Emilia Fridman. Stability of linear descriptor systems with delay: a Lyapunov-based approach. J. Math. Anal. Appl., 273(1):24 – 44, 2002.

[7] W. Michiels. Spectrum-based stability analysis and stabilisation of systems described by delay differential algebraic equations. IET Control Theory Appl., 5(16):1829–1842, 2011.

[8] L. F. Shampine and P. Gahinet. Delay-differential-algebraic equations in control theory. Appl. Numer. Math., 56(3-4):574–588, March 2006.

[9] H. Tian, Q. Yu, and J. Kuang. Asymptotic stability of linear neutral delay differential-algebraic equations and Runge–Kutta methods. SIAM J. Numer. Anal., 52(1):68–82, 2014.

[10] Phi Ha. Spectral characterizations of solvability and stability for delay differential-algebraic equations. Accepted for Acta Math. Vietnamica, url: https://arxiv.org/abs/1802.01148, 2018.

[11] J.K. Hale and S.M.V. Lunel. Introduction to Functional Differential Equations. Springer, 1993.

[12] P. Kunkel and V. Mehrmann. Differential-Algebraic Equations – Analysis and Numerical Solution. EMS Publishing House, Zu¨rich, Switzerland, 2006.

[13] E. Hairer, C. Lubich, and M. Roche. The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods. Springer-Verlag, Berlin, Germany, 1989.

[14] H. Radjavi and P. Rosenthal. Simultaneous Trianqularizability. Universitext, Springer, New york, 2000.

[15] Richard Bellman and Kenneth L. Cooke. Differential-difference equations. Mathematics in Science and Engineering. Elsevier Science, 1963.

[16] E. Jarlebring. The Lambert W function and the spectrum of some multi- dimensional time-delay systems. Automatica, 43(12):2124 – 2128, 2007.

Mathematics Subject Classification (2010): 34A09, 34A12, 65L05, 65H10.

*Keywords:*Differential-algebraic equation · Time delay · Exponential stability · Weak stability · Simultaneous triangularizable

## References

[1] S. L. Campbell. Nonregular 2D descriptor delay systems. IMA J. Math. Control Appl., 12:57–67, 1995.

[2] Nguyen Huu Du, Vu Hoang Linh, Volker Mehrmann, and Do Duc Thuan. Stability and robust stability of linear time-invariant delay differential- algebraic equations. SIAM J. Matr. Anal. Appl., 34(4):1631–1654, 2013.

[3] Phi Ha and Volker Mehrmann. Analysis and reformulation of linear delay differential-algebraic equations. Electr. J. Lin. Alg., 23:703–730, 2012.

[4] Phi Ha and Volker Mehrmann. Analysis and numerical solution of linear delay differential-algebraic equations. BIT, 56:633 – 657, 2016.

[5] S. L. Campbell and V. H. Linh. Stability criteria for differential-algebraic equations with multiple delays and their numerical solutions. Appl. Math Comput., 208(2):397 – 415, 2009.

[6] Emilia Fridman. Stability of linear descriptor systems with delay: a Lyapunov-based approach. J. Math. Anal. Appl., 273(1):24 – 44, 2002.

[7] W. Michiels. Spectrum-based stability analysis and stabilisation of systems described by delay differential algebraic equations. IET Control Theory Appl., 5(16):1829–1842, 2011.

[8] L. F. Shampine and P. Gahinet. Delay-differential-algebraic equations in control theory. Appl. Numer. Math., 56(3-4):574–588, March 2006.

[9] H. Tian, Q. Yu, and J. Kuang. Asymptotic stability of linear neutral delay differential-algebraic equations and Runge–Kutta methods. SIAM J. Numer. Anal., 52(1):68–82, 2014.

[10] Phi Ha. Spectral characterizations of solvability and stability for delay differential-algebraic equations. Accepted for Acta Math. Vietnamica, url: https://arxiv.org/abs/1802.01148, 2018.

[11] J.K. Hale and S.M.V. Lunel. Introduction to Functional Differential Equations. Springer, 1993.

[12] P. Kunkel and V. Mehrmann. Differential-Algebraic Equations – Analysis and Numerical Solution. EMS Publishing House, Zu¨rich, Switzerland, 2006.

[13] E. Hairer, C. Lubich, and M. Roche. The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods. Springer-Verlag, Berlin, Germany, 1989.

[14] H. Radjavi and P. Rosenthal. Simultaneous Trianqularizability. Universitext, Springer, New york, 2000.

[15] Richard Bellman and Kenneth L. Cooke. Differential-difference equations. Mathematics in Science and Engineering. Elsevier Science, 1963.

[16] E. Jarlebring. The Lambert W function and the spectrum of some multi- dimensional time-delay systems. Automatica, 43(12):2124 – 2128, 2007.