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 () is proposed. Then, we characterize the in term of a spectral condition for some special classes of DDAEs. | VNU Journal of Science: Mathematics – Physics, Vol. 34, No. 2 (2018) 52-64 On the Stability Analysis of Delay Differential-Algebraic Equations Ha Phi* VNU University of Science, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam Received 13 April 2018 Revised 28 May 2018; Accepted 14 July 2018 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 () is proposed. Then, we characterize the in term of a spectral condition for some special classes of DDAEs. Keywords: Differential-algebraic equation, time delay, exponential stability, weak stability, simultaneous triangularizable. Mathematics Subject Classification (2010): 34A09, 34A12, 65L05, 65H10. 1. Introduction ∗ 0F Our focus in the present paper is on the stability analysis of linear homogeneous, constant coefficients delay differential-algebraic equations (DDAEs) of the following form E x (= t ) A x(t ) + B x(t − τ ), for all t ∈ [0, ∞), (1) where E , A, B ∈ n ,n , x :[−τ , ∞) → n ,τ > 0 is a constant delay. DDAEs of the form (1) can be considered as a general combination of two important classes of dynamical systems, namely differentialalgebraic equations (DAEs) = E x (t ) A x(t ), for all t ∈ [0, ∞), (2) where the matrix E is allowed to be singular (det E = 0), and delay-differential equations (DDEs) x (= t ) A x(t ) + B x(t − τ ), for all t ∈ [0, ∞), (3) _ ∗ Tel.: 84-963304784. Email: https// 52 H. Phi / VNU Journal of Science: Mathematics – Physics, Vol. 34, No. 2 (2018) 52-64 53 Due to the presence of both differential and difference operators, as well as the algebraic constraints, the study for DDAEs is much more .
