### Bolh-Perron Theorem for Differential Algebraic Equations

## Main Article Content

## Abstract

**Abstract. **This paper is concerned with the Bolh-Perron theorem for differential algebraic equations. We prove that the system

E(t)x'(t)=B(t)x(t), t > t_{0}

is exponentially stable if and only if for any bounded input *q*, the equation

E(t)x'(t)=B(t)x(t)+q(t), t > t_{0}

has a bounded solution.

*Keywords:*Differential-algebraic equation, input-output bounded function, asymptotic stability.

## References

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[2] M. Bracke, On stability radii of parametrized linear differential-algebraic systems, Ph.D. Thesis, University of Kaiserslautern, 2000.

[3] N.H.Du, V.H. Linh, Robust stability of implicit linear systems containing a small parameter in the leading term, IMA J. Mathematical Control Information (in press).

[4] E. Griepentrog, R. M¨arz, Differential-algebraic equations and their numerical treatment, Teubner-TextezurMathematik, Leibzig 1986.

[5] R. M¨arz (1998), ”Extra-ordinary differential equation attempts to an analysis of differential algebraic system”, Progress in Mathematics, 168, pp. 313-334.

[6] L. Qiu, E.J. Davison, The stability robustness of generalized eigenvalues, IEEE Transactions on Automatic Control, 37(1992), 886-891.

[7] Du, N.H.; Tien, L.H., On the Exponential Stability of Dynamic Equations on Time Scales, J. Math. Anal. Appl. 331(2007), pp. 1159–1174.

[8] Linh, Vu Hoang; Nga, Ngo Thi Thanh Bohl-Perron type stability theorems for linear singular difference equations. Vietnam J. Math. 46 (2018), no. 3, 437–451. 39A06 (39A30)

[9] N.H.Du and N. C. Liem; Linear transformations and Floquet theorem for linear implicit dynamic equations on time scales, Asian-European Journal of Mathematics 6,No. 1(2013).