## 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
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