Surveys in Mathematics and its Applications

ISSN1842-6298 (electronic), 1843-7265 (print)

Volume 9 (2014), 1 -- 54## EXPLICIT STABILITY CONDITIONS FOR NEUTRAL TYPE VECTOR FUNCTIONAL DIFFERENTIAL EQUATIONS. A SURVEY

## Michael I. Gil'

Abstract. This paper is a survey of the recent results of the author on the stability of linear and nonlinear neutral type functional differential equations. Mainly, vector equations are considered. In particular, equations whose nonlinearities are causal mappings are investigated. These equations include neutral type, ordinary differential, differential-delay, integro-differential and other traditional equations. Explicit conditions for the Lyapunov, exponential, input-to-state and absolute stabilities are derived. Moreover, solution estimates for the considered equations are established. They provide us the bounds for the regions of attraction of steady states. A part of the paper is devoted to the Aizerman type problem from the the absolute stability theory. The main methodology presented in the paper is based on a combined usage of the recent norm estimates for matrix-valued functions with the generalized Bohl - Perron principle, positivity conditions for fundamental solutions of scalar equations and properties of the so called generalized norm2010 Mathematics Subject Classification:34K20; 34K99; 93D05; 93D25.

Keywords:functional differential equations; neutral type equations; linear and nonlinear equations; exponential stability; absolute stability; L^{2}-stability, input-to-state stability, causal mappings; Bohl - Perron principle; Aizerman problem.

## References

R. Agarwal, L. Berezansky, E. Braverman and A. Domoshnitsky,

Nonoscillation Theory of Functional Differential Equations and Applications, Elsevier, Amsterdam, 2012. MR2908263. Zbl 1253.34002.M. A. Aizerman,

On a conjecture from absolute stability theory, Ushekhi Matematicheskich Nauk ,4(4), (1949) 187-188. In Russian.N. V. Azbelev and P.M. Simonov,

Stability of Differential Equations with Aftereffects, Stability Control Theory Methods Appl.20, Taylor & Francis, London, 2003. MR1965019. Zbl 1049.34090.L. Berezansky and E. Braverman,

On exponential stability of a linear delay differential equation with an oscillating coefficient, Appl. Math. Lett.,22(2009), no. 12, 1833-1837. MR2558549. Zbl 1187.34096.L. Berezansky, E. Braverman and A. Domoshnitsky,

Stability of the second order delay differential equations with a damping term, Differ. Equ. Dyn. Syst.16(2008), no. 3, 185-205. MR2473987. Zbl 1180.34077.C. Corduneanu,

Functional Equations with Causal Operators, Taylor and Francis, London, 2002. MR1949578. Zbl 1042.34094.L. Yu. Daleckii and M. G. Krein,

Stability of Solutions of Differential Equations in Banach Space, Amer. Math. Soc., Providence, R. I. 1971. MR0352639.N. Dunford and J.T. Schwartz,

Linear Operators, part I, Interscience Publishers, Inc., New York, 1966. MR1009162.A. Feintuch and R. Saeks,

System Theory. A Hilbert Space Approach.Ac. Press, New York, 1982. MR0663906. Zbl 0488.93003.M. I. Gil',

On one class of absolutely stable systems, Soviet Physics Doklady,280(4), (1983) 811-815. MR0747002.M. I. Gil',

Stability of Finite and Infinite Dimensional Systems, Kluwer, N. Y, 1998 MR1666431.M. I. Gil',

On Aizerman-Myshkis problem for systems with delay, Automatica,36, (2000) 1669-1673 MR1831724. Zbl 0980.93061.M. I. Gil',

On bounded input-bounded output stability of nonlinear retarded systems, Robust and Nonlinear Control,10, (2000), 1337-1344. MR1801524.M. I. Gil',

Boundedness of solutions of nonlinear differential delay equations with positive Green functions and the Aizerman - Myshkis problem, Nonlinear Analysis, TMA,49, (2002) 1065-1068. MR1942666.M. I. Gil',

Operator Functions and Localization of Spectra, Lecture Notes in Mathematics, Vol.1830, Springer-Verlag, Berlin, 2003. MR2032257. Zbl 1032.47001.M. I. Gil',

Absolute and input-to-state stabilities of nonautonomous systems with causal mappings, Dynamic Systems and Applications,18, (2009) 655-666 MR2562294.M. I. Gil',

L, Internat. J. Robust Nonlinear Control,^{2}-absolute and input-to-state stabilities of equations with nonlinear causal mappings19(2009), no. 2, 151--167. MR2482231. Zbl 1242.34120.M. I. Gil',

The L, Int. J. Dynamical Systems and Differential Equations,^{p}- version of the generalized Bohl - Perron principle for vector equations with delay3, no. 4 (2011) 448-458. MR2911980. Zbl 06238329.M. I. Gil',

Stability of vector functional differential equations: a survey, Quaestiones Mathematicae,35(2012), 1 - 49. MR2931307.M. I. Gil',

Exponential stability of nonlinear neutral type systems, Archives Control Sci.,22 (LVIII), (2012) , no 2, 125-143. MR3088452.M. I. Gil',

Estimates for fundamental solutions of neutral type functional differential equations, Int. J. Dynamical Systems and Differential Equations,4(2012) no. 4, 255-273 MR2988915.M. I. Gil',

The generalized Bohl-Perron principle for the neutral type vector functional differential equations, Mathematics of Control, Signals, and Systems (MCSS),25(1)(2013), 133-145 MR3022296.M. I. Gil',

Stability of Vector Differential Delay Equations, Birkhäuser Verlag, Basel, 2013. MR3026099. Zbl 1272.34003.M. I. Gil',

On Aizerman's type problem for neutral type systems, European Journal of Control,19, no. 2 (2013), 113-117. Zbl 1293.93626.M. I. Gil',

Input-to-State Stability of Neutral Type Systems, Discussiones Mathematicae, Differential Inclusions, Control and Optimization,33 (1) (2013), 1-12. MR3136579. Zbl 06238329.A. Halanay,

Differential Equations: Stability, Oscillation, Time Lags, Academic Press, New York, 1966 MR0216103. Zbl 0144.08701.J. K. Hale and S. M. V. Lunel,

Introduction to Functional Differential Equations, Springer-Verlag, New-York, 1993. MR1243878. Zbl 0787.34002.V. Kolmanovskii and A. Myshkis,

Applied Theory of Functional Differential Equations, Kluwer, Dordrecht, 1999. MR1680144.V. B. Kolmanovskii and V. R. Nosov,

Stability of Functional Differential Equations, Ac Press, London, 1986. Zbl 0593.34070. MR0860947. Zbl 0126.02404.M. A. Krasnosel'skij, J. Lifshits and A. Sobolev,

Positive Linear Systems. The Method of Positive Operators, Heldermann Verlag, Berlin, 1989. MR1038527. Zbl 0674.47036.M. A. Krasnosel'skii and P. P. Zabreiko,

Geometrical Methods of Nonlinear analysis, Springer-Verlag, Berlin, 1984. MR0736839.S. G. Krein,

Linear Differential Equations in a Banach Space, Transl. Mathem. Monogr, vol29, Amer. Math. Soc., 1971. MR0342804.V. Lakshmikantham, S. Leela, Z. Drici and F. A. McRae,

Theory of Causal Differential Equations, Atlantis Studies in Mathematics for Engineering and Science,5Atlantis Press, Paris , 2009. MR2596883. Zbl 1219.34002.V. Lupulescu,

Existence of solutions for nonconvex functional differential inclusions, Electron. J. Differential Equations (2004), no. 141, 6 pp. MR2108912. Zbl 1075.34055.V. Lupulescu,

Causal functional differential equations in Banach spaces, Nonlinear Anal.,69(2008), no. 12, 4787–-4795. MR2467270. Zbl 1176.34093.M. Marcus and H. Minc,

A Survey of Matrix Theory and Matrix Inequalities, Allyn and Bacon, Boston, 1964. MR0158896. Zbl 0126.02404.D. S. Mitrinovic, J. E. Pecaric and A.M. Fink,

Inequalities Involving Functions and their Integrals and Derivatives, Kluwer Academic Publishers, Dordrecht, 1991. MR1190927. Zbl 0744.26011.A. D. Myshkis,

On some problems of theory of differential equations with deviation argument, Uspechi Matemat. Nauk,194, (1977) no 2, 173-202. In Russian.S. I. Niculescu,

Delay Effects on Stability: A Robust Control Approach, Lecture Notes ins Control and Information Sciences,269, Springer-Verlag, London, 2001. Zbl 0997.93001.A. M. Ostrowski,

Note on bounds for determinants with dominant principal diagonals, Proc. of AMS,3, (1952) 26-30. MR0052380. Zbl 0046.01203.D. Popescu, V. Rasvan and R. Stefan,

Applications of stability criteria to time delay systems, Electronic Journal of Qualitative Theory of Differential Equations, Proc. 7th Coll. QTDE, no.18, (2004), 1-20. MR2170486.V. Rasvan,

Absolute Stability of Equations with Delay, Nauka, Moscow, 1983. In Russian.V. Rasvan,

Delay independent and delay dependent Aizerman problem, in Open Problem Book (V. D. Blondel and A. Megretski eds.) pp. 102-107, 15th Int'l Symp. on Math. Theory Networks and Systems MTNS15, Univ. Notre Dame USA, August 12-16, 2002.A. A. Voronov,

Systems with a differentiable nondecreasing nonlinearity that are absolutely stable in the Hurwitz angle, Dokl. Akad. Nauk SSSR,234, (1977) no. 1, 38-41. In Russian. Zbl 0382.93046.

Michael I. Gil'

Department of Mathematics, Ben Gurion University of the Negev

P.0. Box 653, Beer-Sheva 84105, Israel.

e-mail: gilmi@bezeqint.net