Abstract
The solvability and stability analyses of linear time invariant systems of delay differential-algebraic equations (DDAEs) are analyzed. The behavior approach is applied to DDAEs in order to establish characterizations of their solvability in terms of spectral conditions. Furthermore, examples are delivered to demonstrate that the eigenvalue-based approach in analyzing the exponential stability of dynamical systems is only valid for a special class of DDAEs, namely, non-advanced. Then, a new concept of weak stability is proposed and studied for DDAEs whose matrix coefficients pairwise commute.
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Ascher, U.M., Petzold, L.R.: The numerical solution of delay-differential algebraic equations of retarded and neutral type. SIAM J. Numer. Anal. 32(5), 1635–1657 (1995)
Baker, C.T.H., Paul, C.A.H., Tian, H.: Differential algebraic equations with after-effect. J. Comput. Appl. Math. 140(1–2), 63–80 (2002)
Bellen, A., Zennaro, M.: Numerical Methods for Delay Differential Equations. Oxford University Press, New York (2003)
Bellman, R., Cooke, K.L.: Differential-Difference Equations. Academic Press, New York-London (1963)
Brenan, K.E., Campbell, S.L., Petzold, L.R.: Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, 2nd edn. SIAM, Philadelphia (1996)
Byers, R., Geerts, T., Mehrmann, V.: Descriptor systems without controllability at infinity. SIAM J. Control Optim. 35(2), 462–479 (1997)
Byers, R., Kunkel, P., Mehrmann, V.: Regularization of linear descriptor systems with variable coefficients. SIAM J. Control Optim. 35(1), 117–133 (1997)
Campbell, S.L.: Singular linear systems of differential equations with delays. Appl. Anal. 11(2), 129–136 (1980)
Campbell, S.L.: Nonregular 2D descriptor delay systems. IMA J. Math. Control Inform. 12, 57–67 (1995)
Campbell, S.L., Linh, V.H.: Stability criteria for differential-algebraic equations with multiple delays and their numerical solutions. Appl. Math. Comput. 208(2), 397–415 (2009)
Corless, R.M., Gonnet, G.H., Hare, D.E.G., Jeffrey, D.J., Knuth, D.E.: On the Lambert W function. Adv. Comput. Math. 5(4), 329–359 (1996)
Du, N.H., Linh, V.H., Mehrmann, V., Thuan, D.D.: Stability and robust stability of linear time-invariant delay differential-algebraic equations. SIAM J. Matrix Anal. Appl. 34(4), 1631–1654 (2013)
Fridman, E.: Stability of linear descriptor systems with delay: a Lyapunov-based approach. J. Math. Anal. Appl. 273(1), 24–44 (2002)
Gluesing-Luerssen, H.: Linear Delay-Differential Systems with Commensurate Delays: an Algebraic Approach Lecture Notes in Mathematics, vol. 1770. Springer, Berlin (2002)
Gohberg, I., Lancaster, P., Rodman, L.: Matrix Polynomials. Academic Press, New York-London (1982)
Guglielmi, N., Hairer, E.: Computing breaking points in implicit delay differential equations. Adv. Comput. Math. 29(3), 229–247 (2008)
Ha, P.: Analysis and Numerical Solutions of Delay Differential-Algebraic Equations. Dissertation, Institut für Mathematik, TU Berlin, Berlin, Germany (2015)
Ha, P., Mehrmann, V.: Analysis and reformulation of linear delay differential-algebraic equations. Electr. J. Linear Algebra 23, 703–730 (2012)
Ha, P., Mehrmann, V.: Analysis and numerical solution of linear delay differential-algebraic equations. BIT 56(2), 633–657 (2016)
Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential-Equations. Springer, New York (1993)
Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1990)
Khusainov, D.Y., Shuklin, G.V.: Linear autonomous time-delay system with permutation matrices solving. Stud. Univ. Zilina Math. Ser. 17(1), 101–108 (2003)
Kunkel, P., Mehrmann, V.: Differential-Algebraic Equations. Analysis and Numerical Solution. EMS, Zürich (2006)
Linh, V.H., Thuan, D.D.: Spectrum-based robust stability analysis of linear delay differential-algebraic equations. In: Numerical Algebra, Matrix Theory, Differential-Algebraic Equations and Control Theory, Festschrift in Honor of Volker Mehrmann. Chapter 19, pp. 533–557. Springer, Cham (2015)
Michiels, W.: Spectrum-based stability analysis and stabilisation of systems described by delay differential algebraic equations. IET Control Theory Appl. 5(16), 1829–1842 (2011)
Motzkin, T.S., Taussky, O.: Pairs of matrices with property. L. Trans. Am. Math. Soc. 73, 108–114 (1952)
Polderman, J.W., Willems, J.C.: Introduction to Mathematical Systems Theory. a Behavioural Approach. Springer, New York (1998)
Pospisil, M.: Representation and stability of solutions of systems of functional differential equations with multiple delays. Electron. J. Qual. Theory Differ. Equ. 54, 30 (2012)
Shampine, L.F., Gahinet, P.: Delay-differential-algebraic equations in control theory. Appl. Numer. Math. 56(3–4), 574–588 (2006)
Stuart, A.M., Humphries, A.R.: Dynamical Systems and Numerical Analysis. Cambridge University Press, Cambridge (1996)
Tian, H., Yu, Q., Kuang, J.: Asymptotic stability of linear neutral delay differential-algebraic equations and Runge–Kutta methods. SIAM J. Numer. Anal. 52 (1), 68–82 (2014)
Trenn, S.: Regularity of distributional differential algebraic equations. Math. Control Signals Syst. 21(3), 229–264 (2009)
Zhu, W., Petzold, L.R.: Asymptotic stability of linear delay differential-algebraic equations and numerical methods. Appl. Numer. Math. 24(2–3), 247–264 (1997)
Acknowledgements
The author would like to thank the anonymous referee for his constructive comments and suggestions that improve the quality of this paper. The author also thanks Stephan Trenn for helpful comments and fruitful discussions on the first topic of this article.
Funding
This research is funded by the Vietnamese National Foundation for Science and Technology Development (NAFOSTED) under the project number 101.01-2017.302.
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Ha, P. Spectral Characterizations of Solvability and Stability for Delay Differential-Algebraic Equations. Acta Math Vietnam 43, 715–735 (2018). https://doi.org/10.1007/s40306-018-0279-7
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DOI: https://doi.org/10.1007/s40306-018-0279-7
Keywords
- Differential-algebraic equations
- Time delay
- Matrix polynomials
- Commutative
- Exponential stability
- Weak stability