Exponential stability for systems of delay differential equations with block matrices
Section snippets
Introduction and preliminaries
Block matrices are widely used in various real-world models, see, for example, [1], [2], [3], [4], [5]. Properties of such matrices can be found in [6], [7], [8].
Stability conditions for systems of ordinary differential equations (ODEs) with a block matrix were obtained in [9], [10], [11], [12], [13], [14], [15], [16]. The advantage of the monograph [17] is that it contains both properties of block matrices and asymptotic stability conditions for linear delay differential
The main theorem
To state the main result of the paper, we introduce a matrix with positive entries
Theorem 1 Let there exist such that , , be defined as in (9) and let at least one of the following equivalent conditions hold (a) is an -matrix, (b) the spectral radius of satisfies . Then system (2) is uniformly exponentially stable.
Proof To apply Lemma 1, we explore boundedness of
Examples
Example 1 Consider system (2), where is a 10 × 10 tridiagonal block matrix, , ,
, . We have for the norm , Then for matrix in Theorem 1 we have: Calculations of the spectral radius on MATLAB give , hence by Theorem 1 system (2) is uniformly exponentially stable. Compare
Conclusion
We give easily verifiable, especially with an application of accessible computer tools, explicit exponential stability conditions for linear delay differential systems with block matrices. Such delay differential systems are quite common in various applications, motivating this study. In Example 1 we illustrated uniform exponential stability of the system , where is a 10 × 10 block matrix. It is interesting that two other exponential stability tests, where the block
Acknowledgments
The second author acknowledges the support of NSERC, Canada , the grant RGPIN-2020-03934. The authors are grateful to the anonymous reviewers whose thoughtful comments contributed to the quality of presentation.
References (25)
- et al.
Input-to-state stability of switched systems with explicit gain functions
Systems Control Lett.
(2018) Sign properties of Metzler matrices with applications
Linear Algebra Appl.
(2017)- et al.
On the stability of large matrices
J. Comput. Appl. Math.
(1999) - et al.
Simple uniform exponential stability conditions for a system of linear delay differential equations
Appl. Math. Comput.
(2015) - et al.
Stability of simultaneously block triangularisable switched systems with partial state reset
Internat. J. Control
(2017) A note on J-positive block operator matrices
Integral Equations Operator Theory
(2015)- et al.
Computer-Controlled Systems with Delay: A Transfer Function Approach
(2019) - et al.
Symmetry-independent stability analysis of synchronization patterns
SIAM Rev.
(2020) - et al.
Stability of the essential spectrum of the diagonally and off-diagonally dominant block matrix linear relations
J. Pseudo-Differ. Oper. Appl.
(2016) - et al.
A note on recursive Schur complements, block Hurwitz stability of Metzler matrices, and related results
IEEE Trans. Automat. Control
(2017)
Stability of solutions of linear systems with block-diagonal reflecting matrices
Vestn. Beloruss. Gos. Univ. Ser. 1 Fiz. Mat. Inform.
Sufficient conditions for asymptotic liapunov stability of special system of linear differential equations with constant coefficients
Int. J. Gen. Syst.
Cited by (5)
D-stability of the model of the Stieltjes string related to the functional differential equations
2022, Examples and CounterexamplesD-stability of the model of the Stieltjes string
2023, Applicable AnalysisEXPONENTIAL STABILITY FOR SYSTEMS OF VECTOR NEUTRAL DIFFERENTIAL EQUATIONS
2022, Functional Differential EquationsBifurcations of a Fractional-Order Four-Neuron Recurrent Neural Network with Multiple Delays
2022, Computational Intelligence and NeuroscienceNew and improved criteria on qualitative results for functional-differential systems
2022, Nonlinear Studies