# 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)
$H_{\infty}$ analysis of nonlinear stochastic time-delay systems. (English) Zbl 1153.93355
Summary: The $H_{\infty}$ analysis problem is studied for a general class of nonlinear stochastic systems with time-delay. The stochastic systems are described in terms of stochastic functional differential equations. The Razumikhin-type lemma is employed to establish sufficient conditions for the time-delay stochastic systems to be internally stable, and the $H_{\infty}$ analysis problem is studied in order to quantify the disturbance rejection attenuation level of the nonlinear stochastic time-delay system. In particular, the paper obtains the general conditions under which the $L_2$ gain of the system is less than or equal to a given constant. Some easy-to-test criteria are also given so as to determine whether the nonlinear stochastic time-delay system under investigation is internally stable and whether it achieves certain $H_{\infty}$ performance index. Finally, illustrative examples are provided to show the usefulness of the proposed theory.

##### MSC:
 93B36 $H^\infty$-control 93C23 Systems governed by functional-differential equations 93E15 Stochastic stability
Full Text:
##### References:
 [1] Abdallah, S. H.: Stability and persistence in plankton models with distributed delays. Chaos, solitons & fractals 17, No. 5, 879-884 (2003) · Zbl 1033.92032 [2] Arnold, L.: Stochastic differential equations: theory & applications. (1972) · Zbl 0216.45001 [3] Berman N, Shaked U, H\infty for nonlinear stochastic systems. In: Proc 42nd IEEE conference on decision and control. vol. 5, 2003. p. 5025-30 [4] Chiarella, C.; Szidarovszky, F.: Bounded continuously distributed delays in dynamic oligopolies. Chaos, solitons & fractals 18, No. 5, 977-993 (2003) · Zbl 1068.91005 [5] Doyle, J. C.; Glover, K.; Khargonekar, P. P.; Francis, B. A.: State-space solutions to the standard H2 and H$\infty$control problems. IEEE trans automat control 34, 831-847 (1989) · Zbl 0698.93031 [6] El-Gohary, A.; Yassen, M. T.: Optimal control and synchronization of Lotka-Volterra model. Chaos, solitons & fractals 12, No. 11, 2087-2093 (2001) · Zbl 1004.92037 [7] Friedman, A.: Stochastic differential equations and their applications. 2 (1976) · Zbl 0323.60057 [8] Hale, J. K.: Theory of functional differential equations. (1977) · Zbl 0352.34001 [9] He, J. H.: Mysterious pi and a possible link to DNA sequencing. Int J nonlinear sci numer simul 5, No. 3, 263-274 (2004) [10] Helton, J. W.; James, M. R.: Extending H$\infty$control to nonlinear systems: control of nonlinear systems to achieve performance objectives. SIAM adv des contr ser (1999) · Zbl 0941.93003 [11] Hinrichsen, D.; Pritchard, A. J.: Stochastic h\infty. SIAM J contr optim. 36, 1504-1538 (1998) · Zbl 0914.93019 [12] Isidori, A.; Kang, W.: H$\infty$control via measurement feedback for general nonlinear systems. IEEE trans automat contr 40, 466-472 (1995) · Zbl 0822.93029 [13] Kuzmenkov, L. S.; Maximov, S. G.; Guardado, J. L.: On the asymptotic solutions of the coupled quasiparticle-oscillator system. Chaos, solitons & fractals 15, No. 4, 597-610 (2003) · Zbl 1098.81039 [14] Mao, X.: Stability of stochastic differential equations with respect to semi-martingales. (1991) · Zbl 0724.60059 [15] Mao, X.: Exponential stability of stochastic differential equations. (1994) · Zbl 0806.60044 [16] Mao, X.: Razumikhin-type theorems on exponential stability of stochastic functional differential equations. Stoch process appl 5, No. 2, 233-250 (1996) · Zbl 0889.60062 [17] Mao, X.: Razumikhin-type theorems on exponential stability of neutral stochastic functional differential equations. SIAM J math anal 28, No. 2, 389-401 (1997) · Zbl 0876.60047 [18] Mao, X.; Sabanis, S.: Numerical solutions of stochastic differential delay equations under local Lipschitz condition. J comput appl math 51, 215-227 (2003) · Zbl 1015.65002 [19] Raouf, A. F. E.; Moatimid, G. M.: Nonlinear dynamics and stability of two streaming magnetic fluids. Chaos, solitons & fractals 12, No. 7, 1207-1216 (2001) · Zbl 1021.76018 [20] Teel, A. R.: Connection between razumikhin-type theorems and the ISS nonlinear small gain theorem. IEEE trans automat contr 43, 960-964 (1998) · Zbl 0952.93121 [21] Van Der Schaft, A. J.: L2-gain analysis of nonlinear systems and nonlinear control. IEEE trans automat contr 37, No. 6, 770-784 (1992) · Zbl 0755.93037 [22] Van Der Schaft, A. J.: L2-gain and passivity techniques in nonlinear control. Communication and control engineering (2000) · Zbl 0937.93020 [23] Wang, Z.; Ho, D. W. C.: Filtering on nonlinear time-delay stochastic systems. Automatica 39, No. 1, 101-109 (2003) · Zbl 1010.93099 [24] Wang, Z.; Goodall, D. P.; Burnham, K.: On designing observers for time-delay systems with nonlinear disturbances. Int J contr 75, No. 11, 803-811 (2002) · Zbl 1027.93007 [25] Wang, Z.; Qiao, H.; Burnham, K.: On stabilization of bilinear uncertain time-delay stochastic systems with Markovian jumping parameters. IEEE trans automat contr 47, No. 4, 640-646 (2002) [26] Wang, Z.; Huang, B.; Burnham, K.: Stochastic reliable control of a class of uncertain time-delay systems with unknown nonlinearities. IEEE trans circ syst--part I 48, No. 5, 646-650 (2001) · Zbl 1023.93070 [27] Wang, Z.; Huang, B.; Unbehauen, H.: Robust H$\infty$observer design of linear state delayed systems with parametric uncertainty: the discrete-time case. Automatica 35, No. 6, 1161-1167 (1999) · Zbl 1041.93514 [28] Zames, G.: Feedback and optimal sensitivity: model reference transformations, multiplicative seminorms, and approximate inverses. IEEE trans automat contr 26, 301-320 (1981) · Zbl 0474.93025