# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Asymptotic and exponential stability of certain third-order non-linear delayed differential equations: frequency domain method. (English) Zbl 1200.34081

The asymptotic stability of the zero solution for some third order delayed differential equations is studied by the frequency domain method. Also, for the non-homogeneous equations, the existence of exponentially stable bounded solutions is proved. The known theorems on absolute stability for the system of the form

$\stackrel{˙}{X}\left(t\right)=AX\left(t\right)+BX\left(t-\tau \right)-Q\varphi \left(\sigma \left(t\right)\right)+P\left(t\right),\phantom{\rule{1.em}{0ex}}\sigma \left(t\right)={C}_{1}X\left(t\right)+{C}_{2}X\left(t\right)$

with sector conditions for $\varphi$ are used to obtain the results.

##### MSC:
 34K20 Stability theory of functional-differential equations
##### References:
 [1] Afuwape, A. U.: Conditions on the behaviour of solutions for third-order nonlinear differential equations, An. stii ale univ. ’Al.i.cuza’, iasi 30, 30-34 (1984) · Zbl 0625.34049 [2] Afuwape, A. U.: Frequency-domain approach to nonlinear oscillations of some third-order differential equations, J. nonlinear analysis, TMA 10, No. 12, 1459-1470 (1986) · Zbl 0612.34050 · doi:10.1016/0362-546X(86)90115-X [3] Afuwape, A. U.: Frequency-domain approach to some third-order nonlinear differential equations, nonlinear analysis, theory, Methods and applications 71, No. 12, e972-e978 (2009) [4] Afuwape, A. U.: Frequency-domain approach to certain third-order nonlinear delayed differential equations, Directions in mathematics, 49-58 (1999) [5] Afuwape, A. U.: Convergence of solutions of some third order systems of nonlinear ordinary differential equations, An. stii ale univ.’al.i.cuza’ iasi .a. Mat. tomul LV 1, 11-20 (2009) · Zbl 1199.34247 [6] Afuwape, A. U.; Omeike, M. O.: On the stability and boundedness of solutions of a kind of third order delay differential equations, Appl. math. Comput. 200, No. 1, 444-451 (2008) [7] A.U. Afuwape, J. Mawhin, F. Zanolin, An existence theorem for periodic solutions and applications to some nonlinear ordinary and delay equations, preprint, ICTP, Italy. [8] I. Barbalat, Conditions pour un bon Comportment de certaines equations differentielles du troisieme et du Quatrieme ordre, equations differentielles et functionelles non-lineaires, in: P. Janssens, J. Mawhin, N. Rouche (Eds.), (1973) 79 – 91. · Zbl 0296.34024 [9] Barbalat, I.; Halanay, A.: Nouvelles applications de la method frequentielle dans la theorie des oscillations, Rev. roum. Sci. techn. Electrotechn. energ. 16, No. 2, 689-702 (1971) [10] Barbalat, I.; Halanay, A.: Conditions de comportement ’presque lineaire’ dans la theorie des oscillations, Rev. roum. Sci. techn. Electrotechn. energ. 29, No. 2, 321-341 (1974) [11] Brockett, R. W.; Willems, J. L.: Frequency domain stability criteria, Trans. automat. Control 2, No. 1 (1965) [12] Cahlon, B.; Schmidt, D.: Stability criteria for certain third-order delay differential equations, J. comp. Appl. math 188, 319-335 (2006) · Zbl 1094.34051 · doi:10.1016/j.cam.2005.04.034 [13] Duan, Zhiheng; Wang, Jinzhi; Yang, Ying; Huang, Lin: Frequency-domain and time domain methods for feedback nonlinear systems and applications to chaos control, Chaos, solitons and fractals 40, 848-861 (2009) · Zbl 1197.93133 · doi:10.1016/j.chaos.2007.08.034 [14] Gopalsamy, K.: Stability and oscillations in delay differential equation of population dynamics, (1992) · Zbl 0752.34039 [15] Gromova, P. S.; Pelevina, A. F.: Absolute stability of automatic control systems with time-lag, Differential eqs. 13, No. 2, 954-960 (1977) · Zbl 0395.93028 [16] Halanay, A.: Periodic solutions of linear systems with lag, Rev. de math. Pure et appliquee. Acad. R.S.R. 6, No. 1, 141-158 (1961) [17] Halanay, A.: Differential equations: stability, oscillations, time lags, (1966) · Zbl 0144.08701 [18] Halanay, A.: Invariant manifolds for systems with time-lag, Differential equations and dynamical systems, 199-213 (1967) · Zbl 0162.40001 [19] Halanay, A.: Almost periodic solutions for a class of non-linear systems with time lag, Rev. roum. Pures et appliquee 14, No. 9, 1269-1276 (1969) · Zbl 0197.42203 [20] A. Halanay, Systemes a retard. Application des methodes frequentielles Equations differentielles et functionelles non-lineaires ed. P. Janssens, J. Mawhin et N. Rouche, 1973, pp. 357 – 380. · Zbl 0347.34056 [21] Kalman, R. E.: Lyapunov functions for the problem of lurie in automatic control, Proc. national acad. Sci. USA 49, 201-205 (1963) · Zbl 0113.07701 · doi:10.1073/pnas.49.2.201 [22] Leonov, G. A.; Ponomarenko, D. V.; Smirnova, V. B.: Frequency-domain methods for nonlinear analysis, theory and applications, (1996) [23] Popov, V. M.; Halanay, A.: On the stability of non-linear automatic control systems with lagging argument, Aut. rem. Control, 783-786 (1962) · Zbl 0114.04602 [24] Rasvan, Vl.: Absolute stability of a class of non-linear time-lag control systems: I. The fundamental case; II. Critical cases, Rev. roum. Sci. techn. Ser. electrotechn. Et energ. 17, No. 4, 667-681 (1972) [25] Vl. Rasvan, Frequency domain stability criteria for time lag control systems with several non-linearities. I. General results; II. Applications; III. Linear and non-linear systems with monotonic time varying gain, Rev. Roum. Sci. Techn. ser. Electrotechn. et Energ. 17 (2) 369 – 381 [26] Sadek, A. I.: Stability and boundedness of a kind of third order delay differential system, Applied math. Letters, 657-662 (2003) · Zbl 1056.34078 · doi:10.1016/S0893-9659(03)00063-6 [27] Tejumola, H. O.: Existence of periodic solutions of certain third-order non-linear differential equations with delay I, J. nigerian math. Soc. 7, 59-66 (1988) [28] H.O. Tejumola, Resonant and nonresonant oscillations for certain third-order differential equations with delay, in: Joseph Wiener, Jack K. Hale (Eds.), Pitman Research Notes in Mathematics, vol. 272: Ordinary and Delay Differential Equations, 1992, pp. 238 – 243. · Zbl 0792.34069 [29] Tejumola, H. O.; Tchegnani, B.: Stability, boundedness and existence of periodic solutions of some third and fourth order nonlinear delay differential equations, J. nigerian math. Soc. 19, 9-19 (2000) [30] Cemil; Tunc: New results about stability and boundedness of solutions of certain non-linear third-order delay differential equations, Arab. J. Sci. eng. 31, No. 2A, 185-196 (2006) · Zbl 1195.34113 [31] Cemil; Tunc: Stability and boundedness of solutions of nonlinear equations of third-order with delay, Differential equations and control processes 3, 1-13 (2007) [32] Cemil; Tunc: On asymptotic stability of solutions to third order nonlinear differential equations with retarded argument, Commun. appl. Anal. 11, No. 3 – 4, 515-527 (2007) · Zbl 1139.34054 [33] Cemil; Tunc: On the boundedness of solutions of third-order delay differential equations, Differ. equ. (Differ. Ravn.) 44, No. 4, 464-472 (2008) [34] Cemil; Tunc: On the stability and boundedness of solutions to third order nonlinear differential equations with retarded arguments, Nonlinear dynam. 57, No. 1 – 2, 97-106 (2009) [35] Cemil; Tunc: A new boundedness result to nonlinear differential equations of third order with finite lag, Commun. appl. Anal. 13, No. 1, 1-10 (2009) [36] Cemil; Tunc: Stability criteria for certain third order nonlinear delay differential equations, Port. math. 66, No. 1, 71-80 (2009) · Zbl 1166.34329 · doi:10.4171/PM/1831 · doi:http://www.ems-ph.org/journals/show_pdf.php?issn=0032-5155&vol=66&iss=1&rank=5 [37] Cemil; Tunc: On the boundedness of delay differential equations of third order, Arab. J. Sci. eng. Sect. A sci. 34, No. 1, 227-237 (2009) · Zbl 1195.34102 [38] Cemil; Tunc: On the stability and boundedness of solutions of nonlinear vector differential equations of third order, Nonlinear anal. 34, No. 6, 2232-2236 (2009) · Zbl 1162.34043 · doi:10.1016/j.na.2008.03.002 [39] Cemil; Tunc: On the qualitative behaviors of solutions of a kind of nonlinear third order differential equations with a retarded argument, An. stiit. Univ. ’ovidius’ constanta ser. Mat. 17, No. 2, 215-230 (2009) · Zbl 1199.34392 [40] Cemil; Tunc: Bounded solutions to nonlinear delay differential equations of third order, Topol. methods nonlinear anal. 34, No. 1, 131-139 (2009) · Zbl 1187.34088 [41] Yacubovich, V. A.: The matrix method in the theory of the stability of non linear control systems, Aut. rem. Control 25, 905-916 (1964)