# Input-to-state stability of neutral type systems

• Volume: 33, Issue: 1, page 5-16
• ISSN: 1509-9407

top

## Abstract

top
We consider the system $ẋ\left(t\right)-\int {₀}^{\eta }dR̃\left(\tau \right)ẋ\left(t-\tau \right)={\int }_{0}^{\eta }dR\left(\tau \right)x\left(t-\tau \right)+\left[Fx\right]\left(t\right)+u\left(t\right)$ (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.

## How to cite

top

Michael I. Gil'. "Input-to-state stability of neutral type systems." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 33.1 (2013): 5-16. <http://eudml.org/doc/270674>.

@article{MichaelI2013,
abstract = {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.},
author = {Michael I. Gil'},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {neutral type systems; causal mappings; input-to-state stability},
language = {eng},
number = {1},
pages = {5-16},
title = {Input-to-state stability of neutral type systems},
url = {http://eudml.org/doc/270674},
volume = {33},
year = {2013},
}

TY - JOUR
AU - Michael I. Gil'
TI - Input-to-state stability of neutral type systems
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2013
VL - 33
IS - 1
SP - 5
EP - 16
AB - 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.
LA - eng
KW - neutral type systems; causal mappings; input-to-state stability
UR - http://eudml.org/doc/270674
ER -

## References

top
1. [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 Zbl1011.93101
2. [2] C. Corduneanu, Functional Equations with Causal Operators, Taylor and Francis, London, 2002.
3. [3] A. Feintuch and R. Saeks, System Theory. A Hilbert Space Approach, Ac. Press, New York, 1982.
4. [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 Zbl0796.93004
5. [5] M.I. Gil', Stability of Finite and Infinite Dimensional Systems, Kluwer, N.Y, 1998 doi: 10.1007/978-1-4615-5575-9
6. [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 Zbl0979.93107
7. [7] M.I. Gil', Operator Functions and Localization of Spectra, Lecture Notes in Mathematics, Vol. 1830, Springer-Verlag, Berlin, 2003. doi: 10.1007/b93845 Zbl1032.47001
8. [8] M.I. Gil', Absolute and input-to-state stabilities of nonautonomous systems with causal mappings, Dynamic Systems and Applications 18 (2009) 655-666. Zbl1185.34103
9. [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 Zbl1188.34096
10. [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. [11] M.I. Gil', Exponential stability of nonlinear neutral type systems, Archives of Control Sci 22 (2) (2012), 125-143. Zbl1270.93095
12. [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 Zbl1263.34114
13. [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 Zbl1097.93027
14. [14] V.B. Kolmanovskii and V.R. Nosov, Stability of Functional Differential Equations, Academic Press, London, 1986.
15. [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 Zbl1005.93044
16. [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 Zbl1001.93072
17. [17] A.M. Ostrowski, Note on bounds for determinants with dominant principal diagonals, Proc. of AMS 3 (1952), 26-30. Zbl0046.01203
18. [18] J. Partingtona and C. Bonnetb, ${H}^{\infty }$ 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. [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 Zbl1168.34356
20. [20] E.D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems, Springer-Verlag, New York, 1990. Zbl0703.93001

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.