Analysis of Linear Variable Coefficient Delay Differential-Algebraic Equations
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The analysis of general linear variable coefficient delay differential-algebraic systems (DDAEs) is presented. The solvability for DDAEs is investigated and a reformulation procedure to regularize a given DDAE is developed. Based on this regularization procedure existence and uniqueness of solutions and consistency of initial functions is analyzed as well as other structural properties of DDAEs like smoothness requirements. We also present some examples to demonstrate that for the numerical solution of a DDAE, a reformulation of the system before applying numerical methods is essential.
KeywordsDifferential-algebraic equation Delay differential-algebraic equation Regularization Existence of solutions Uniqueness of solutions Consistency conditions
Mathematics Subject Classification34A09 34A12 65L05 65H10
We thank Vinh Tho Ma for carrying the numerical simulations in Examples 5.1 and 5.2. Phi Ha has been supported by Deutsche Forschungsgemeinschaft through Sonderforschungsbereich 910 Control of self-organizing nonlinear systems: Theoretical methods and application concepts. V. Mehrmann and A. Steinbrecher has been supported by European Research Council through Advanced Grant MODSIMCONMP.
- 1.Ascher, U.M., Petzold, L.R.: Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM Publications, Philadelphia (1998)Google Scholar
- 3.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 (Mar. 2002)Google Scholar
- 12.Gear, C. W.: Simulation: conflicts between real-time and software. In Mathematical Software III, Proceedings of the Symposium at the University of Wisconsin, March 28–30, 1977, (ed. J.R. Rice), pp. 121–138. Academic Press, New York (1977)Google Scholar
- 18.Heeb, H., Ruehli, A.: Retarded models for PC board interconnects-or how the speed of light affects your spice circuit simulation. In IEEE International Conference on Computer-Aided Design, 1991. ICCAD-91, pp. 70–73 (1991)Google Scholar
- 24.Misra, V., Gong, W.B., Towsley, D.: Fluid-based analysis of a network of AQM routers supporting TCP flows with an application to red. SIGCOMM Comput. Commun. Rev. 30(4), 151–160 (Aug. 2000)Google Scholar
- 27.Shampine, L.F., Gahinet, P.: Delay-differential-algebraic equations in control theory. Appl. Numer. Math. 56(3–4), 574–588 (Mar. 2006)Google Scholar
- 28.Steinbrecher, A.: Regularization of quasi-linear differential-algebraic equations in multibody dynamics. In Proceedings of Multibody Dynamics 2011-ECCOMAS Thematic Conference Brussels (Bruxelles, Belgium, July 4–7, 2011) (2011)Google Scholar
- 29.Wim, M., Silviu-Iulian, N.: Stability and Stabilization of Time-Delay Systems: An Eigenvalue-Based Approach. SIAM, Philadelphia (2007)Google Scholar