Abstract
In this paper, the asymptotic stabilization of linear distributed parameter control systems of neutral type is considered. Specifically, we study control systems described by a special type of abstract neutral functional differential equation with finite delay. Assuming appropriate conditions, and using the spectral properties of quasi-compact semigroups, we show that the usual spectral controllability assumption implies the feedback stabilization of the system. We applied our results to the stabilization of several real systems of first and second order described by partial neutral functional differential equations.
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References
Adimy M, Ezzinbi K (1999) Existence and linearized stability for partial neutral functional differential equations with nondense domains. Differ Equ Dyn Syst 7:371–417
Adimy M, Ezzinbi K, Laklach M (2001) Spectral decomposition for partial neutral functional differential equations. Can Appl Math Q 9:1–34
Adimy M, Elazzouzi A, Ezzinbi K (2007) Bohr-Neugebauer type theorem for some partial neutral functional differential equations. Nonlinear Anal 66:1145–1160
Alia M, Ezzinbi K, Fatajou S (2009) Exponential dichotomy and pseudo almost automorphy for partial neutral functional differential equations. Nonlinear Anal 71:2210–2226
Ammari K, Nicaise S, Pignotti C (2010) Feedback boundary stabilization of wave equations with interior delay. Syst Control Lett 59:623–628
Balakrishnan AV (1981) Applied functional analysis, 2nd edn. Springer, New York
Baser U, Kizilsac B (2007) Dynamic output feedback \(H_{\infty }\) control problem for linear neutral systems. IEEE Trans Autom Contr 52:1113–1118
Benchimol CD (1978) Feedback stabilizability in Hilbert spaces. Appl Math Optim 4:225–248
Bounit H, Hadd S (2006) Regular linear systems governed by neutral FDEs. J Math Anal Appl 320:836–858
Byrnes CI, Spong MW, Tam T-J (1984) A several complex variables approach to feedback stabilization of linear neutral delay-differential systems. Math Syst Theory 17:97–133
Curtain RF, Pritchard AJ (1978) Infinite dimensional linear systems theory. Lect Notes in Control and Inform Sci 8, Springer, Berlin
Curtain RF, Zwart HJ (1995) An introduction to infinite-dimensional linear systems theory. Springer, New York
Datko R, Lagness J, Poilis MP (1986) An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J Control Optim 24:152–156
Datko R (1988) Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks. SIAM J Control Optim 26:697–713
Diagana T, Hernández E, Dos Santos JP (2009) Existence of asymptotically almost automorphic solutions to some abstract partial neutral integro-differential equations. Nonlinear Anal 71:248–257
Dusser X, Rabah R (2001) On exponential stabilizability for linear neutral systems. Math Probl Eng 7:67–86
Elharfi A, Bounit H, Hadd S (2006) Representation of infinite-dimensional neutral non-autonomous control systems. J Math Anal Appl 323:497–512
Engel K-J, Nagel R (2000) One-parameter semigroups for linear evolution equations. Springer, New York
Ezzinbi K, Fatajou S, N’Guérékata G (2009) Pseudo-almost-automorphic solutions to some neutral partial functional differential equations in Banach spaces. Nonlinear Anal 70:1641–1647
Fridman E, Shaked U (2002) A descriptor systems approach to \(H_{\infty }\) control of linear time-delay systems. IEEE Trans Autom Control 47:253–270
Fridman E, Nicaise S, Valein J (2010) Stabilization of second order evolution equations with unbounded feedback with time-dependent delay. SIAM J Control Optim 48:5028–5052
Goldberg S (1985) Unbounded linear operator. Dover, New York
Hadd S, Zhong Q-C (2007) Conditions on feedback stabilization of systems with state and input delays in Banach spaces. In: Proceedings 46th IEEE conf. decision and, control, pp 2094–2099
Hadd S (2008) Singular functional differential equations of neutral type in Banach spaces. J Funct Anal 254:2069–2091
Hadd S, Rhandi A (2008) Feedback theory for neutral equations in infinite dimensional state spaces. Note Mat 28:43–68
Hale J (1977) Theory of functional differential equations. Springer, New York
Hale J, Verduyn Lunel SM (1993) Introduction to functional differential equations. Springer, New York
Hale J (1994) Partial neutral functional differential equations. Rev Roumaine Math Pures Appl 39:339–344
Hale J (1994) Coupled oscillators on a circle. Resenhas do Instituto de Matemática e Estatística da Universidade de São Paulo 1:441–457
Hale JK, Verduyn Lunel SM (2002) Strong stabilization of neutral functional differential equations. IMA J Math Control 19:5–23
Hale JK, Verduyn Lunel SM (2003) Stability and control of feedback systems with time delays. Int J Syst Sci 34:497–504
Han Q-L (2009) Improved stability criteria and controller design for linear neutral systems. Automatica 45:1948–1952
Han Z-J, Xu GQ (2011) Exponential stability of Timoshenko beam system with delay terms in boundary feedbacks. ESAIM Control Optim Calc Var 17:552–574
Hao Z-H, Hu G-D, Li H-D (2010) Controller based on spectral decomposition for neutral delay systems. Acta Autom Sin 36:615–619
Hautus MLJ (1969) Controllability and observability conditions of linear autonomous systems. Indag Math 31:443–448
Henríquez HR (1985) On non-exact controllable systems. Int J Control 42:71–83
Henríquez H R (2001) Stabilization of hereditary distributed parameter control systems. Syst Control Lett 44: 35–43
Henríquez HR (2008) Approximate controllability of linear distributed control systems. Appl Math Lett 21:1041–1045
Kato T (1995) Perturbation theory for linear operators. Springer, Berlin
Kolmanovskii V, Myshkis A (1992) Applied theory of functional differential equations. Kluwer, Dordrecht
Logemann H, Pandolfi L (1994) A note on stability and stabilizability of neutral systems. IEEE Trans Autom Control 39:138–143
Logemann H, Rebarber R, Weiss G (1996) Conditions for robustness and nonrobustness of the stability of feedback systems with respect to small delays in the feedback loop. SIAM J Control Optim 34:572–600
Marchenko VM, Yakimenko AA (2007) Stabilizing controller design for delay systems of neutral type. Differ Equ 43:1516–1523
Martin RH (1987) Nonlinear operators and differential equations in Banach spaces. Robert E. Krieger Publishing Company, Florida
Michiels W, Vyhlídal T (2005) An eigenvalue based approach for the stabilization of linear time-delay systems of neutral type. Automatica 41:991–998
Murray JD (2002) Mathematical biology. I. An introduction, Third edn. Springer, Berlin
Nakagiri S-I, Yamamoto M (2001) Feedback stabilization of linear retarded systems in Banach spaces. J Math Anal Appl 262:160–178
Nicaise S, Pignotti C (2006) Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM J Control Optim 45:1561–1585
Nicaise S, Valein J (2010) Stabilization of second order evolution equations with unbounded feedback with delay. ESAIM Control Optim Calc Var 16:420–456
O’Connor DA, Tarn TJ (1983) On stabilization by state feedback for neutral differential difference equations. IEEE Trans Autom Control 28:615–618
O’Reilly J (1975) Observers for linear systems. Academic Press, London
Pandolfi L (1975) Feedback stabilization of functional differential equations. Boll Un Mat Ital (4) 11: Suppl fasc 3, 626–635
Pandolfi L (1976) Stabilization of neutral functional differential equations. J Optim Theory Appl 20:191–204
Park JuH (2004) Delay-dependent guaranteed cost stabilization criterion for neutral delay-differential systems: matrix inequality approach. Comput Math Appl 47:1507–1515
Pazy A (1983) Semigroups of linear operators and applications to partial differential equations. Springer, New York
Pritchard AJ, Zabczyk J (1981) Stability and stabilizability of infinite dimensional systems. SIAM Rev 23:25–52
Qiu J, He H, Shi P (2011) Robust stochastic stabilization and \(H_{\infty }\) control for neutral stochastic systems with distributed delays. Circuits Syst Signal Process 30:287–301
Rabah R, Sklyar GM (2005) On a class of strongly stabilizable systems of neutral type. Appl Math Lett 18:463–469
Rabah R, Sklyar GM, Rezounenko AV (2008) On strong regular stabilizability for linear neutral type systems. J Differ Equ 245:569–593
Rebarber R, Townley S (1999) Robustness with respect to delays for exponential stability of distributed parameter systems. SIAM J Control Optim 37:230–244
Salamon D (1984) Control and observation of neutral systems. Pitman, Boston
Salsa S (2008) Partial differential equations in action. From modelling to theory. Springer, Milan
Sun J, Liu GP, Chen J (2009) Delay-dependent stability and stabilization of neutral time-delay systems. Int J Robust Nonlinear Control 19:1364–1375
Taylor AE (1958) Introduction to functional analysis. Wiley, New York
Triggiani R (1975) On the stabilizability problem in Banach space. J Math Anal Appl 52:383–403
Wonham WM (1979) Linear multivariable control: a geometric approach. Springer, Berlin
Wu J, Xia H (1996) Self-sustained oscillations in a ring array of coupled lossless transmission lines. J Differ Equ 124:247–278
Wu J, Xia H (1999) Rotating waves in neutral partial functional differential equations. J Dynam Differ Equ 11:209–238
Wu J (1996) Theory and applications of partial functional differential equations. Springer, New York
Xia X, Liu K (2010) Spectral properties and finite pole assignment of linear neutral systems in Banach spaces. Abstract Appl Anal Article ID 948764. doi:10.1155/2010/948764
Xiang Z, Sun Y-N, Chen Q (2011) Robust reliable stabilization of uncertain switched neutral systems with delayed switching. Appl Math Comput 217:9835–9844
Xu GQ, Yung SP, Li LK (2006) Stabilization of wave systems with input delay in the boundary control. ESAIM Control Optim Calc Var 12:770–785
Yamamoto M (1987) On the stabilization of evolution equations by feedback with time-delay: an operator-theoretical approach. J Fac Sci Univ Tokyo Sect IA Math 34:165–191
Zhou S, Zhou L (2010) Improved exponential stability criteria and stabilisation of T-S model-based neutral systems. IET Control Theory Appl 4:2993–3002
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This research was supported in part by CONICYT under Grant FONDECYT No. 1090009.
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Zamorano, S., Henríquez, H.R. Feedback stabilization of abstract neutral linear control systems. Math. Control Signals Syst. 25, 345–386 (2013). https://doi.org/10.1007/s00498-012-0103-1
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DOI: https://doi.org/10.1007/s00498-012-0103-1