Input-to-state stability of neutral type systems

• Autorzy: Michael I. Gil'
• Języki publikacji: EN

Abstrakty

EN

We consider the system
$ẋ(t) - ∫₀^{η} dR̃(τ) ẋ(t-τ) = ∫_0^{η} dR(τ)x(t-τ) + [Fx](t) + u(t)$
(ẋ(t) ≡ dx(t)/dt), where x(t) is the state, u(t) is the input, R(τ),R̃(τ) are matrix-valued functions, and F is a causal (Volterra) mapping. Such equations enable us to consider various classes of systems from the unified point of view. Explicit input-to-state stability conditions in terms of the L²-norm are derived. Our main tool is the norm estimates for the matrix resolvents, as well as estimates for fundamental solutions of the linear parts of the considered systems, and the Ostrowski inequality for determinants.

Słowa kluczowe

EN

neutral type systems; causal mappings; input-to-state stability

Kategorie tematyczne

• 93D25: Input-output approaches
• 34K40: Neutral equations

• Wydawca: Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra
• Czasopismo: Discussiones Mathematicae, Differential Inclusions, Control and Optimization
• Rocznik: 2013
• Tom: 33
• Numer: 1
• Strony: 5-16

Daty

wydano

2013

otrzymano

2012-01-08

Twórcy

autor

Michael I. Gil'

• Department of Mathematics, Ben Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel

Bibliografia

• [1] M. Arcak and A. Teel, Input-to-state stability for a class of Lur'e systems, Automatica 38 (11) (2002), 1945-1949. doi: 10.1016/S0005-1098(02)00100-0
• [2] C. Corduneanu, Functional Equations with Causal Operators, Taylor and Francis, London, 2002.
• [3] A. Feintuch and R. Saeks, System Theory. A Hilbert Space Approach, Ac. Press, New York, 1982.
• [4] T.T. Georgiou and M.C. Smith, Graphs, causality, and stabilizability: linear, shift-invariant systems on L²[0,8), Math. Control Signals Systems 6 (1993), 195-223. doi: 10.1007/BF01211620
• [5] M.I. Gil', Stability of Finite and Infinite Dimensional Systems, Kluwer, N.Y, 1998 doi: 10.1007/978-1-4615-5575-9
• [6] M.I. Gil', On bounded input-bounded output stability of nonlinear retarded systems, Robust and Nonlinear Control 10 (2000), 1337-1344. doi: 10.1002/1099-1239(20001230)10:15<1337::AID-RNC543>3.0.CO;2-B
• [7] M.I. Gil', Operator Functions and Localization of Spectra, Lecture Notes in Mathematics, Vol. 1830, Springer-Verlag, Berlin, 2003. doi: 10.1007/b93845
• [8] M.I. Gil', Absolute and input-to-state stabilities of nonautonomous systems with causal mappings, Dynamic Systems and Applications 18 (2009) 655-666.
• [9] M.I. Gil', L²-absolute and input-to-state stabilities of equations with nonlinear causal mappings, Internat. J. Robust Nonlinear Control 19 (2) (2009), 151-167. doi: 10.1002/rnc.1305
• [10] M.I. Gil', Stability of vector functional differential equations: a survey, Quaestiones Mathematicae 35 (2012), 1-49. doi: 10.2989/16073606.2012.671261
• [11] M.I. Gil', Exponential stability of nonlinear neutral type systems, Archives of Control Sci 22 (2) (2012), 125-143.
• [12] M.I. Gil', Estimates for fundamental solutions of neutral type functional differential equations, Int. J. Dynamical Systems and Differential Equations 4 (4) (2012), 255-273. doi: 10.1504/IJDSDE.2012.049904
• [13] V.L. Kharitonov, Lyapunov functionals and Lyapunov matrices for neutral type time delay systems: a single delay case, International Journal of Control 78 (2005), 783-800. doi: 10.1080/00207170500164837
• [14] V.B. Kolmanovskii and V.R. Nosov, Stability of Functional Differential Equations, Academic Press, London, 1986.
• [15] M. Krichman, E.D. Sontag and Y. Wang, Input-output-to-state stability, SIAM J. Control Optimization 39 (6) (2000), 1874-1928. doi: 10.1137/S0363012999365352
• [16] J.-J. Loiseau, M. Cardelli and X. Dusser, Neutral-type time-delay systems that are not formally stable are not BIBO stabilizable, IMA J. Math. Control Inform. 19 (2002), 217-227. doi: 10.1093/imamci/19.1_and_2.217
• [17] A.M. Ostrowski, Note on bounds for determinants with dominant principal diagonals, Proc. of AMS 3 (1952), 26-30.
• [18] J. Partingtona and C. Bonnetb, $H^∞$ and BIBO stabilization of delay systems of neutral type, Systems & Control Letters 52 (2004), 283-288. doi: 10.1016/j.sysconle.2003.09.014
• [19] R. Rakkiyappan and P. Balasubramaniam, LMI conditions for global asymptotic stability results for neutral-type neural networks with distributed time delays, Appl. Math. and Comput. 204 (2008), 317-324. doi: 10.1016/j.amc.2008.06.049
• [20] E.D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems, Springer-Verlag, New York, 1990.

Identyfikatory

DOI

10.7151/dmdico1143