Exponential stability for systems of delay differential equations with block matrices

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Abstract

We obtain efficient exponential stability tests for a system ẋ(t)=A(t)x(h(t)), where A is a block matrix and h(t) is a delay function, in terms of norms and matrix measures of blocks. Compared to the analysis of the whole matrix A, handling blocks can be more manageable even for the system ẋ(t)=A(t)x(t) without delay. In a presented example, the criterion applied to A fails, while considering appropriate blocks leads to exponential stability. Another example illustrates efficiency even in the case of a non-delay system.

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) ẋ(t)=A(t)x(t)with a block matrix A(t) 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 D=(dij)i,j=1m with positive entries dii=τAiiμ(Aii)[t0,)Aii[t0,),dij=τAiiμ(Aii)[t0,)Aij[t0,)+Aijμ(Aii)[t0,),ji.

Theorem 1

Let there exist αi such that μ(Aii(t))αi<0, i=1,,m, D be defined as in (9) and let at least one of the following equivalent conditions hold

(a) B=ED is an M-matrix,

(b) the spectral radius of D satisfies r(D)<1.

Then system (2) is uniformly exponentially stable.

Proof

To apply Lemma 1, we explore boundedness of

Examples

Example 1

Consider system (2), where A(t) is a 10 × 10 tridiagonal block matrix, d=2, m=5, Aii=40.5sin2t0.5sin2t0.5cos2t40.5cos2t,Aij=a0cos2itsin2(it)sin2itcos2(it),ij,|ij|=1, i=1,,5, th(t)τ=0.09,a0=0.65. We have for the norm , μ(Aii)=4,Aii=5,Aij=0.65,ij,|ij|=1.Then for matrix D in Theorem 1 we have: dii=0.5625,dij=0.2356,ij,|ij|=1,dij=0,|ij|>1.Calculations of the spectral radius on MATLAB give r(D)=0.9706<1, 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 ẋ(t)=A(t)x(h(t)), where A(t) 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)

  • Al’sevichL.A.

    Stability of solutions of linear systems with block-diagonal reflecting matrices

    Vestn. Beloruss. Gos. Univ. Ser. 1 Fiz. Mat. Inform.

    (2002)
  • AmrahovS.G.

    Sufficient conditions for asymptotic liapunov stability of special system of linear differential equations with constant coefficients

    Int. J. Gen. Syst.

    (2002)
  • Cited by (5)

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