Exact null controllability, complete stabilizability and continuous final observability of neutral type systems

Rabah Rabah; Grigory Sklyar; Pavel Barkhayev

doi:10.1515/amcs-2017-0034

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Exact null controllability, complete stabilizability and continuous final observability of neutral type systems

International Journal of Applied Mathematics and Computer Science

Volume 27 (2017): Issue 3 (September 2017)

By: Rabah Rabah, Grigory Sklyar and Pavel Barkhayev

Open Access

|Sep 2017

Abstract

For abstract linear systems in Hilbert spaces we revisit the problems of exact controllability and complete stabilizability (stabilizability with an arbitrary decay rate), the latter property being related to exact null controllability. We also consider the case when the feedback is not bounded. We obtain a characterization of complete stabilizability for neutral type systems. Conditions for exact null controllability of neutral type systems are discussed. By duality, we obtain a result about continuous final observability. Illustrative examples are given.

References

Curtain, R.F., Logemann, H., Townley, S. and Zwart, H. (1997). Well-posedness, stabilizability, and admissibility for Pritchard–Salamon systems, Journal of Mathematical Systems, Estimation, and Control4(4): 493–496.
Search in Google Scholar Back to article
Curtain, R.F. and Zwart, H. (1995). An Introduction to Infinitedimensional Linear Systems Theory, Texts in Applied Mathematics, Vol. 21, Springer-Verlag, New York, NY.
Search in Google Scholar Back to article
Douglas, R.G. (1966). On majorization, factorization, and range inclusion of operators on Hilbert space, Proceedings of the American Mathematical Society17: 413–415.10.1090/S0002-9939-1966-0203464-1
Open DOI Search in Google Scholar Back to article
Dusser, X. and Rabah, R. (2001). On exponential stabilizability of linear neutral systems, Mathematical Problems in Engineering7(1): 67–86.10.1155/S1024123X01001533
Open DOI Search in Google Scholar Back to article
Guo, F., Zhang, Q. and Huang, F. (2003). Well-posedness and admissible stabilizability for Pritchard–Salamon systems, Applied Mathematical Letters16(1): 65–70.10.1016/S0893-9659(02)00145-3
Open DOI Search in Google Scholar Back to article
Hale, J.K. and Verduyn Lunel, S.M. (1993). Introduction to Functional-Differential Equations, Applied Mathematical Sciences, Vol. 99, Springer-Verlag, New York, NY.
Search in Google Scholar Back to article
Hale, J.K. and Verduyn Lunel, S.M. (2002). Strong stabilization of neutral functional differential equations, IMA Journal of Mathematical Control and Information19(1–2): 5–23.10.1093/imamci/19.1_and_2.5
Open DOI Search in Google Scholar Back to article
Ito, K. and Tarn, T.J. (1985). A linear quadratic optimal control for neutral systems, Nonlinear Analysis9(7): 699–727.10.1016/0362-546X(85)90013-6
Search in Google Scholar Back to article
Khartovskiĭ, V.E. and Pavlovskaya, A.T. (2013) Complete controllability and controllability for linear autonomous systems of neutral type, Automation and Remote Control79(5): 769–784.10.1134/S0005117913050032
Open DOI Search in Google Scholar Back to article
Kuperman, L.M. and Repin, Ju. M. (1971) On the question of controllability in infinite-dimensional spaces, Doklady Akademii Nauk SSSR200(3): 767–769, (in Russian). English translation: Soviet Mathematics—Doklady12(5): 1469–1472.
Search in Google Scholar Back to article
Logemann, H. and Pandolfi, L. (1994) A note on stability and stabilizability of neutral systems, IEEE Transactions on Automatic Control39(1): 138–143.10.1109/9.273351
Open DOI Search in Google Scholar Back to article
Louis, J.-C. and Wexler, D. (1983). On exact controllability in Hilbert spaces, Journal of Differential Equations49(2): 258–269.10.1016/0022-0396(83)90014-1
Open DOI Search in Google Scholar Back to article
Marčenko, V.M. (1979). On the controllability of zero function of time lag systems, Problems of Control and Information Theory/Problemy Upravlenia i Teoria Informacii8(5–6): 421–432.
Search in Google Scholar Back to article
Megan, M. (1975). On the stabilizability and controllability of linear dissipative systems in Hilbert space, Seminarul de Ecuatii Functional, Universitatea din Timisoara32: 1–15.
Search in Google Scholar Back to article
Metel'skiĭ, A.V. and Khartovskii, S.A. (2016). Criteria for modal controllability of linear systems of neutral type, Differential Equations52(11): 1453–1468.10.1134/S0012266116110070
Search in Google Scholar Back to article
Metel'skiĭ, A.V. and Minyuk, S.A. (2006). A criterion of constructive identifiability and complete controllability of linear time-independent systems of neutral type, Izvestiya Rossiiskoi Akademii Nauk: Teoriya i Sistemy Upravleniya (5): 15–23, (in Russian). English translation: Journal of Computer and Systems Sciences International45(5): 690–698.
Search in Google Scholar Back to article
Michiels, W. and Niculescu, S.-I. (2007). Stability and Stabilization of Time-Delay Systems. An Eigenvalue-Based Approach, Advances in Design and Control, Vol. 12, SIAM, Philadelphia, PA.10.1137/1.9780898718645
Search in Google Scholar Back to article
O’Connor, D.A. and Tarn, T.J. (1983). On stabilization by state feedback for neutral differential equations, IEEE Transactions on Automatic Control28(5): 615–618.10.1109/TAC.1983.1103286
Open DOI Search in Google Scholar Back to article
Olbrot, A.W. and Pandolfi, L. (1988). Null controllability of a class of functional-differential systems, International Journal on Control47(1): 193–208.10.1080/00207178808906006
Search in Google Scholar Back to article
Pandolfi, L. (1976). Stabilization of neutral functional differential equations, Journal of Optimization Theory and Applications20(2): 191–204.10.1007/BF01767451
Search in Google Scholar Back to article
Pazy, A. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Vol. 44, Springer-Verlag, New York, NY.10.1007/978-1-4612-5561-1_7
Open DOI Search in Google Scholar Back to article
Pritchard, A.J. and Salamon, D. (1987). The linear quadratic control problem for infinite dimensional systems with unbounded input and output operators, SIAM Journal on Control and Optimization25(1): 121–144.10.1137/0325009
Open DOI Search in Google Scholar Back to article
Rabah, R. and Karrakchou, J. (1997). On exact controllability and complete stabilizability for linear systems in Hilbert spaces, Applied Mathematics Letters10(1): pp. 35–40.10.1016/S0893-9659(96)00107-3
Open DOI Search in Google Scholar Back to article
Rabah, R. and Sklyar, G.M. (2007). The analysis of exact controllability of neutral-type systems by the moment problem approach, SIAM Journal on Control and Optimization46(6): 2148–2181.10.1137/060650246
Open DOI Search in Google Scholar Back to article
Rabah, R. and Sklyar, G.M. (2016). Exact observability and controllability for linear neutral type systems, Systems and Control Letters89: 8–15.10.1016/j.sysconle.2015.12.010
Search in Google Scholar Back to article
Rabah, R., Sklyar, G.M. and Barkhayev, P.Y. (2012). Stability and stabilizability of mixed retarded-neutral type systems, ESAIM Control, Optimization and Calculus of Variations18(3): 656–692.10.1051/cocv/2011166
Search in Google Scholar Back to article
Rabah, R., Sklyar, G.M. and Barkhayev, P.Y. (2014). On the exact controllability and observability of neutral type systems, Communications on Mathematical Analysis17(2): 279–294.
Search in Google Scholar Back to article
Rabah, R., Sklyar, G.M. and Barkhayev, P.Y. (2016). On exact controllability of neutral time-delay systems, Ukrainian Mathematical Journal68(6): 800–815.10.1007/s11253-016-1265-7
Search in Google Scholar Back to article
Rabah, R., Sklyar, G.M. and Rezounenko, A.V. (2005). Stability analysis of neutral type systems in Hilbert space, Journal of Differential Equations214(2): 391–428.10.1016/j.jde.2004.08.001
Search in Google Scholar Back to article
Rabah, R., Sklyar, G.M. and Rezounenko, A.V. (2008). On strong regular stabilizability for linear neutral type systems, Journal of Differential Equations245(3): 569–593.10.1016/j.jde.2008.02.041
Search in Google Scholar Back to article
Rhandi, A. (2002). Spectral Theory for Positive Semigroups and Applications, Quaderni di Matematica, Vol. 2, ESE—Salento University Publishing, Lecce.
Search in Google Scholar Back to article
Richard, J.-P. (2003). Time-delay systems: An overview of some recent advances and open problems, Automatica39(10): 1667–1694.10.1016/S0005-1098(03)00167-5
Search in Google Scholar Back to article
Salamon, D. (1983). Neutral functional differential equations and semigroups of operators, in F. Kappel et al. (Eds.), Control Theory for Distributed Parameter Systems and Applications, Lecture Notes in Control and Information Sciences, Vol. 54, Springer, Berlin, pp. 188–207.10.1007/BFb0043949
Open DOI Search in Google Scholar Back to article
Salamon, D. (1984). Control and Observation of Neutral Systems, Research Notes in Mathematics, Vol. 91, Pitman (Advanced Publishing Program), Boston, MA.
Search in Google Scholar Back to article
Salamon, D. (1987). Infinite dimensional linear systems with unbounded control and observation: A functional analytical approach, Transactions of American Mathematical Society300(2): 383–431.10.2307/2000351
Search in Google Scholar Back to article
Sklyar, G.M. and Szkibiel, G. (2013). Controlling a nonhomogeneous Timoshenko beam with the aid of the torque, International Journal of Applied Mathematics and Computer Science23(3): 587–598, DOI: 10.2478/amcs-2013-0044.10.2478/amcs-2013-0044
Open DOI Search in Google Scholar Back to article
Tucsnak, M. and Weiss, G. (2009). Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts, Basel/Birkhäuser, Boston, MA.10.1007/978-3-7643-8994-9
Search in Google Scholar Back to article
Wonham, W.M. (1985). Linear Multivariable Control: A Geometric Approach, 3rd Edn., Springer, New York, NY.
Search in Google Scholar Back to article
Zabczyk, J. (1976). Complete stabilizability implies exact controllability, Seminarul de Ecuatii Functional, Universitatea din Timisoara38: 1–7.
Search in Google Scholar Back to article
Zabczyk, J. (1992). Mathematical Control Theory: An Introduction, Systems & Control: Foundations & Applications, Birkhäuser, Boston, MA.
Search in Google Scholar Back to article
Zeng, Y., Xie, Z. and Guo, F. (2013). On exact controllability and complete stabilizability for linear systems, Applied Mathematical Letters26(7): 766–768.10.1016/j.aml.2013.02.008
Open DOI Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.1515/amcs-2017-0034 | Journal eISSN: 2083-8492 | Journal ISSN: 1641-876X

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Language: English

Page range: 489 - 499

Submitted on: Nov 16, 2016

Accepted on: Jun 7, 2017

Published on: Sep 23, 2017

Published by: University of Zielona Góra

In partnership with: Paradigm Publishing Services

Publication frequency: 4 issues per year

Keywords:

exact null controllability,

complete stabilizability,

Related subjects:

© 2017 Rabah Rabah, Grigory Sklyar, Pavel Barkhayev, published by University of Zielona Góra
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Volume 27 (2017): Issue 3 (September 2017)