Stability conditions for non-autonomous linear differential equations in a Hilbert space via commutators

Year 2018, Volume: 1 Issue: 1, 6 - 11, 30.06.2018

Michael Gil'

Abstract

In a Hilbert space $\mathcal{H}$ we consider the equation $dx(t)/dt=(A+B(t))x(t)$ $(t\ge 0),$ where $A$ is a constant bounded operator, and $B(t)$ is a piece-wise continuous function defined on $[0,\8)$ whose values are bounded operators in $\mathcal{H}$. Conditions for the exponential stability are derived in terms of the commutator $AB(t)-B(t)A$. Applications to integro-differential equations are also discussed. Our results are new even in the finite dimensional case.

Keywords

Hilbert space, Differential equation, Stability, Integro-differential equation, Barbashin type equation

References

• [1] V.M. Aleksandrov, and E.V. Kovalenko, Problems in Continuous Mechanics with Mixed Boundary Conditions, Nauka, Moscow 1986. In Russian.
• [2] J. Appel, A. Kalitvin and P. Zabreiko, Partial Integral Operators and Integrodifferential Equations, Marcel Dekker, New York, 2000.
• [3] J.A.D. Appleby and D.W. Reynolds, On the non-exponential convergence of asymptotically stable solutions of linear scalar Volterra integro-differential equations, Journal of Integral Equations and Applications, 14, no 2 (2002), 521-543.
• [4] K. M. Case, P. F. Zweifel, Linear Transport Theory, Addison-Wesley, Reading Mass. 1967.
• [5] M.C. Cercignani, Mathematical Methods in Kinetic Theory, Macmillian, New York, 1969.
• [6] Chuhu Jin and Jiaowan Luo, Stability of an integro-differential equation, Computers and Mathematics with Applications, 57 (2009) 1080–1088.
• [7] Yu L. Daleckii, and M. G. Krein, Stability of Solutions of Differential Equations in Banach Space, Amer. Math. Soc., Providence, R. I. 1974.
• [8] A. Domoshnitsky and Ya. Goltser, An approach to study bifurcations and stability of integro-differential equations. Math. Comput. Modelling, 36 (2002), 663-678.
• [9] A.D. Drozdov, Explicit stability conditions for integro-differential equations with periodic coefficients, Math. Methods Appl. Sci. 21 (1998), 565-588.
• [10] A.D. Drozdov and M. I. Gil’, Stability of linear integro-differential equations with periodic coefficients. Quart. Appl. Math. 54 (1996), 609-624.
• [11] N.T. Dung, On exponential stability of linear Levin-Nohel integro-differential equations, Journal of Mathematical Physics 56, 022702 (2015); doi:10.1063/1.4906811
• [12] M.I. Gil’, Operator Functions and Localization of Spectra, Lecture Notes In Mathematics vol. 1830, Springer-Verlag, Berlin, 2003.
• [13] M.I. Gil’, Spectrum and resolvent of a partial integral operator. Applicable Analysis, 87, no. 5, (2008) 555–566.
• [14] M.I. Gil’, On stability of linear Barbashin type integro-differential equations, Mathematical Problems in Engineering, 2015, Article ID 962565, (2015), 5 pages.
• [15] M.I. Gil’, A bound for the Hilbert-Schmidt norm of generalized commutators of nonself-adjoint operators, Operators and Matrices, 11, no. 1 (2017), 115-123
• [16] Ya. Goltser and A. Domoshnitsky, Bifurcation and stability of integrodifferential equations, Nonlinear Anal. 47 (2001), 953-967.
• [17] Ya. Coltser and A. Domoshnitsky, About reducing integro-differential equations with infinite limits of integration to systems of ordinary differential equations. Advances in Difference Equations 2013:187, (2013) 17 pages.
• [18] H.G. Kaper, C.G. Lekkerkerker, and J. Hejtmanek, Spectral Methods in Linear Transport Theory, Birkhauser, Basel, 1982.
• [19] B. G. Pachpatte, On a parabolic integrodifferential equation of Barbashin type, Comment. Math.Univ. Carolin. 52, no. 3 (2011) 391-401
• [20] W.J. Rugh, Linear System Theory. Prentice Hall, Upper Saddle River, New Jersey, 1996.
• [21] H. R. Thieme, A differential-integral equation modelling the dynamics of populations with a rank structure, Lect. Notes Biomath. 68 (1986), 496-511
• [22] J. Vanualailai and S. Nakagiri, Stability of a system of Volterra integro-differential equations J. Math. Anal. Appl. 281 (2003) 602-619
• [23] B. Zhang, Construction of Liapunov functionals for linear Volterra integrodifferential equations and stability of delay systems, Electron. J. Qual. Theory Differ. Equ. 30 (2000) 1-17.