×

Stability criteria for differential-algebraic equations with multiple delays and their numerical solutions. (English) Zbl 1169.65079

The authors study the multi-delay differential-algebraic equation (MDDAE) \[ A\dot{x}(t) + Bx(t) + \sum_{i=1}^{M}\left[C_i \dot{x}(t-\tau_i)+ D_i x(t-\tau_i)\right ] = 0, \] where \(A, B, C_i, D_i\) are \((m\times{m})\) constant matrices, \(0<\tau_1<\cdots <\tau_M\), the matrix \(A\) is singular with \(rank A<m\). A special subclass of the equation with delays \(\tau_i=i\tau\) with a given \(\tau>0\) is also under consideration. A sufficient spectral conditions of delay-independent asymptotic stability, as well as some checkable algebraic criteria for the asymptotic stability for analytic solutions are obtained. It is proved that the spectral type stability conditions for the special subclass of the MDDAE provide asymptotic stability of the \(\theta\)-methods and the backward differentiation formula methods. Besides, solvability and stability of a class of weakly regular single delay DAEs are analyzed.

MSC:

65L20 Stability and convergence of numerical methods for ordinary differential equations
34K06 Linear functional-differential equations
34K20 Stability theory of functional-differential equations
34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)
65L80 Numerical methods for differential-algebraic equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Ascher, U.; Petzold, L.R., Stability of computational methods for constrained dynamics systems, SIAM J. sci. comput., 14, 1, 95-120, (1993) · Zbl 0773.65044
[2] Ascher, U.; Petzold, L.R., The numerical solution of delay-differential-algebraic equations of retarded and neutral type, SIAM J. numer. anal., 32, 1635-1657, (1995) · Zbl 0837.65070
[3] Brenan, K.E.; Campbell, S.L.; Petzold, L.R., Numerical solution of initial value problems in differential algebraic equations, (1996), SIAM Philadelphia · Zbl 0844.65058
[4] Byers, R.; Nichols, N., On the stability radius of a generalized state-space system, Linear algebra appl., 188/189, 113-134, (1993) · Zbl 0783.65056
[5] Campbell, S.L., Singular linear systems of differential equations with delays, Appl. anal., 11, 129-136, (1980) · Zbl 0444.34062
[6] S.L. Campbell, 2-D (differential-delay) implicit systems, in: Proc. IMACS World Congress on Sci. Comp., Dublin, 1991, pp. 1828-1829.
[7] Campbell, S.L., Nonregular 2D descriptor delay systems, IMA J. math. cont. inf., 12, 57-67, (1995) · Zbl 0835.34085
[8] Cao, Y.; Li, S.; Petzold, L.R.; Serban, R., Adjoint sensitivity analysis for differential-algebraic equations: the adjoint DAE system and its numerical solution, SIAM J. sci. comput., 24, 1076-1089, (2003) · Zbl 1034.65066
[9] Du, N.H.; Linh, V.H., Robust stability of implicit linear systems containing a small parameter in the leading term, IMA J. math. control inform., 23, 67-84, (2006) · Zbl 1106.93044
[10] Fridman, E., Stability of linear descriptor systems with delay: a Lyapunov-based approach, J. math. anal. appl., 273, 24-44, (2002) · Zbl 1032.34069
[11] Griepentrog, E.; März, R., Differential-algebraic equations and their numerical treatment, (1986), Teubner-Texte zur Mathematik Leibzig · Zbl 0629.65080
[12] Hale, J.K.; Verduyn Lunel, S.M., Introduction to functional equations, (1993), Springer-Verlag · Zbl 0787.34002
[13] Hu, G.D.; Hu, G.D.; Cahlon, B., Algebraic criteria for stability of linear neutral systems with a single delay, J. comput. appl. math., 135, 125-133, (2001) · Zbl 0996.34064
[14] He, P.; Cao, D.Q., Algebraic stability criteria of linear neutral systems with multiple time delays, Appl. math. comput., 155, 643-653, (2004) · Zbl 1060.34044
[15] Kato, T., Perturbation theory for linear operators, (1966), Springer-Verlag New York, NY · Zbl 0148.12601
[16] Kunkel, P.; Mehrmann, V., Differential-algebraic equations. analysis and numerical solution, (2006), EMS Publishing House Zürich, Switzerland · Zbl 0707.65043
[17] Lancaster, P.; Tismenetsky, M., The theory of matrices, (1985), Academic Press, Inc. Orlando, FL · Zbl 0516.15018
[18] Linh, V.H., On the robustness of asymptotic stability for a class of singularly perturbed systems with multiple delays, Acta math. viet., 30, 137-151, (2005) · Zbl 1155.93407
[19] Luzyanina, T.; Roose, D., Periodic solutions of differential algebraic equations with time-delays: computation and stability analysis, J. bifurcation chaos, 16, 67-84, (2006) · Zbl 1106.34045
[20] L. Poppe, The strangeness index of a linear delay differential-algebraic equation of retarded type, in: Proceedings of the Sixth IFAC Workshop on Time-Delay Systems, 2006, 5pp.
[21] Qiu, L.; Davison, E.J., The stability robustness of generalized eigenvalues, IEEE trans. autom. control, 37, 886-891, (1992) · Zbl 0775.93181
[22] Shampine, L.F.; Gahinet, P., Delay-differential-algebraic equations in control theory, Appl. numer. math., 56, 574-588, (2006) · Zbl 1093.93015
[23] Stykel, T., On criteria for asymptotic stability of differential-algebraic equations, Z. angew. math. mech., 92, 147-158, (2002) · Zbl 1014.34037
[24] Xu, S.; Van Dooren, P.; Radu, S.; Lam, J., Robust stability and stabilization for singular systems with state delay and parameter uncertainty, IEEE trans. autom. cont., 47, 1122-1128, (2002) · Zbl 1364.93723
[25] Zhu, W.; Petzold, L.R., Asymptotic stability of linear delay differential-algebraic equations and numerical methods, Appl. numer. math., 24, 247-264, (1997) · Zbl 0879.65060
[26] Zhu, W.; Petzold, L., Asymptotic stability of Hessenberg delay differential-algebraic equations of retarded or neutral type, Appl. numer. math., 27, 309-325, (1998) · Zbl 0937.34059
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.