RESEARCH PAPER
The Practical Feedback Stabilization for Evolution Equations in Banach Spaces
1
Faculty of Sciences of Sfax, Department of Mathematics, University of Sfax, Route Soukra BP1171, 3000 Sfax, Tunisia
Submission date: 2020-11-28
Acceptance date: 2021-05-20
Online publication date: 2021-06-30
Publication date: 2021-06-01
Acta Mechanica et Automatica 2021;15(2):58-65
KEYWORDS
dynamical systems in controllinear operatorcontrollabilityuncertain systemspractical stabilizationBanach spaces
ABSTRACT
This paper investigates the notion of practical feedback stabilization of evolution equations satisfying some relaxed conditions in infinite-dimensional Banach spaces. Moreover, sufficient conditions are presented that guarantee practical stabilizability of uncertain systems based on Lyapunov functions. These results are applied to partial differential equations.
REFERENCES (32)
1.
Chen P. (2021), Periodic solutions to non-autonomous evolution equations with multi-delays, Discrete and Continuous Dynamical Systems, 26(6), 2921–2939.
2.
Chen P., Zhang X., Li Y. (2020a), Cauchy problem for fractional non-autonomous evolution equations, Banach Journal of Mathematical Analysis, 14(2), 559–584.
3.
Chen P., Zhang X., Li Y. (2020b), Existence approximate controllability of fractional evolution equations with nonlocal conditions via resolvent operators, Fractional Calculus Applied Analysis, 23(1), 268–291.
4.
Chen P., Zhang X., Li Y. (2020c), Approximate Controllability of Non-autonomous Evolution System with Nonlocal Conditions, Journal of Dynamical Control Systems, 26(1), 1–16.
5.
Chen P., Zhang X., Li Y. (2021), Cauchy problem for stochastic non-autonomous evolution equations governed by noncompact evolution families, Discrete and Continuous Dynamical Systems, 26(3), 1531–1547.
6.
Curtain R.F., Pritchard A.J. (1978), Infinite Dimensional Linear Systems Theory, Lecture Notes in Control Information Sciences, Springer Verlag, Berlin.
7.
Curtain R.F., Zwart H.J. (1995), An Introduction to Infinite Dimensional Linear Systems Theory, Lecture Notes in Control Information Sciences, Springer Verlag, New York.
8.
Damak H. (2020), Asymptotic stability of a perturbed abstract differential equations in Banach spaces, Operators matrices, 14, 129–138.
9.
Damak H. (2021), Input-to-state stabilty integral input-to-state stability of non-autonomous infinite-dimensional systems, International Journal of Systems Sciences. https://doi.org/10.1080/002077....
10.
Damak H., Ellouze I., Hammami M.A. (2013), A separation principle of a class of time-varying nonlinear systems, Nonlinear Dynamics Systems Theory, 13, 133–143.
11.
Damak H., Hammami M.A. (2016), Stabilization Practical Asymptotic Stability of Abstract Differential Equations, Numerical Functional Analysis Optimization, 37, 1235–1247.
12.
Datko R. (1970), Extending a theorem of A.M. Lyapunov to Hilbert spaces, Journal of Mathematical Analysis Applications, 32, 610–616.
13.
Diesel J., Uhl Jr. J.J (1977), Vector Measures, Mathematical surveys, American Mathematical Society, Rhode Isl.
14.
Dragomir S.S. (2002), Some Gronwall Type Inequalities Applications, School of Communications Informatics, Victoria University of Technology.
15.
Ellouze I. (2019), On the practical separation principle of time-varying perturbed systems, IMA Journal of Mathematical Control Information, 00, 1–16.
16.
Henry D. (1981), Geometric Theory of Semilinear Parabolic Equations of Lecture Notes in Mathematics, Springer Verlag, Berlin.
17.
Ikeda M., Maed H., Kodama S. (1972), Stabilization of linear systems, SIAM Journal on Control, 10, 716–729.
18.
Kalman R.E., Ho Y.C., Narenda K.S. (1963), Controllability of Linear Dynamical Systems, Differential Equations, 1, 189–213.
19.
Kobayashi T. (1989), Feedback stabilization of parabolic distributed parameter systems by discrete-time input-output data, SIAM Journal on Control Optimization, 22, 509–522.
20.
Lakshmikantham V., Leela S., Martynuk A.A. (1998), Practical Stability of Nonlinear Systems, World Scientific, Singapore.
21.
Megan G. (1975), On the stabilizability Controllability of Linear Dissipative Systems in Hilbert spaces, 32 S.E.F, Universitate din Timisoara.
22.
Pazy A. (1983), Semigroups of Linear Operators Applications to Partial Differential Equations, Springer, New York.
23.
Phat V.N. (2001), Stabilization of linear continuous time-varying systems with state delays in Hilbert spaces, Electronic Journal of Differential Equations, 2001, 1–13.
24.
Phat V.N. (2002), New Stabilization criteria for linear time-varying systems with state delays norm-bounded uncertainties, in IEEE Transactions on Automatic Control, 12, 2095–2098.
25.
Phat V.N., Ha Q.P. (2008), New characterization of stabilizability via Riccati equations for LTV systems, IMA Journal of Mathematical Control Information, 25, 419–429.
26.
Phat V.N., Kiet T.T. (2002), On the Lyapunov equation in Banach spaces applications to control problems, International Journal of Mathematics Mathematical Sciences, 29, 155–166.
27.
Teschl G. (2012), Ordinary Differential Equations Dynamicals Systems, Graduate studies in mathematics, American Mathematical Society.
28.
Tsinias J. (1991), Existence of control Lyapunov functions its applications to state feedback stabilizability of nonlinear systems, SIAM Journal on Control Optimization, 29, 457–473.
29.
Wonham W.M. (1967), On Pole assignment in Multi-Input Controlable Linear Systems, IEEE Transactions on Automatic Control, 12(6), 660–665.
30.
Xuejiao H., Zhenchao C. (1999), Controllability of linear systems in non-reflexive Banach spaces, Northeastern Mathematical Journal, 15, 459–464.
32.
Zhoo B. (2017), Stability analysis of nonlinear time-varying systems by Lyapunov functions with indefinite derivatives, IET Control Theory Applications, 11, 1434–1442.
| eISSN: | 2300-5319 |
| ISSN: | 1898-4088 |
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