Exact null controllability, stabilizability, and detectability of linear nonautonomous control systems: a quasisemigroup approach. (English) Zbl 1470.93073

Summary: In the theory control systems, there are many various qualitative control problems that can be considered. In our previous work, we have analyzed the approximate controllability and observability of the nonautonomous Riesz-spectral systems including the nonautonomous Sturm-Liouville systems. As a continuation of the work, we are concerned with the analysis of stability, stabilizability, detectability, exact null controllability, and complete stabilizability of linear non-autonomous control systems in Banach spaces. The used analysis is a quasisemigroup approach. In this paper, the stability is identified by uniform exponential stability of the associated \(C_0\)-quasisemigroup. The results show that, in the linear nonautonomous control systems, there are equivalences among internal stability, stabizability, detectability, and input-output stability. Moreover, in the systems, exact null controllability implies complete stabilizability.


93C25 Control/observation systems in abstract spaces
93B05 Controllability
47D06 One-parameter semigroups and linear evolution equations
Full Text: DOI


[1] Kalman, R. E.; Ho, Y. C.; Narendra, K. S., Controllability of Linear Dynamical Systems, Differrential Equations, 1, 189-213, (1963) · Zbl 0151.13303
[2] Phat, V. N., Weak asymptotic stabilizability of discrete-time systems given by set-valued operators, Journal of Mathematical Analysis and Applications, 202, 2, 363-378, (1996) · Zbl 0858.93055
[3] Phat, V. N., Global stabilization for linear continuous time-varying systems, Applied Mathematics and Computation, 175, 2, 1730-1743, (2006) · Zbl 1131.93044
[4] Phat, V. N.; Ha, Q. P., New characterization of controllability via stabilizability and Riccati equation for LTV systems, IMA Journal of Mathematical Control and Information, 25, 4, 419-429, (2008) · Zbl 1152.93030
[5] Wonham, W. M., On Pole Assignment in Multi-Input Controllable Linear Systems, IEEE Transactions on Automatic Control, AC-12, 6, 660-665, (1967)
[6] Jerbi, H., Asymptotic stabilizability of three-dimensional homogeneous polynomial systems of degree three, Applied Mathematics Letters, 17, 3, 357-366, (2004) · Zbl 1059.93113
[7] Ikeda, M.; Maeda, H.; Kodama, S., Stabilization of linear systems, SIAM Journal on Control and Optimization, 10, 716-729, (1972) · Zbl 0244.93049
[8] Phat, V. N.; Kiet, T. T., On the Lyapunov equation in Banach Spaces and Applications to Control Problems, 29, 155-166, (2002) · Zbl 0999.93034
[9] Guo, F.; Zhang, Q.; Huang, F., Well-posedness and admissible stabilizability for Pritchard-Salamon systems, Applied Mathematics Letters, 16, 1, 65-70, (2003) · Zbl 1019.93027
[10] Rabah, R.; Sklyar, G.; Barkhayev, P., Exact null controllability, complete stabilizability and continuous final observability of neutral type systems, International Journal of Applied Mathematics and Computer Science, 27, 3, 489-499, (2017) · Zbl 1373.93067
[11] Hinrichsen, D.; Pritchard, A. J., Robust stability of linear evolution operators on Banach spaces, SIAM Journal on Control and Optimization, 32, 6, 1503-1541, (1994) · Zbl 0817.93055
[12] Niamsup, P.; Phat, V. N., Linear time-varying systems in Hilbert spaces: exact controllability implies complete stabilizability, Thai Journal of Mathematics, 7, 1, 189-200, (2009) · Zbl 1204.93019
[13] Fu, X.; Zhang, Y., Exact null controllability of non-autonomous functional evolution systems with nonlocal conditions, Acta Mathematica Scientia, 33, 3, 747-757, (2013) · Zbl 1299.34255
[14] Sutrima; Indrati, Ch. R.; Aryati, L., Controllability and Observability of Nonautonomous Riesz-Spectral Systems, Abstract and Applied Analysis, 2018, (2018) · Zbl 1470.93029
[15] Leiva, H.; Barcenas, D., Quasi-semigroups, Evolution Equation and Controllability, Notas de Matematicas, 109, 1-29, (1991)
[16] Sutrima; Indrati, Ch. R.; Aryati, L.; Mardiyana, The fundamental properties of quasi-semigroups, Journal of Physics: Conference Series, 855, 1, (2017)
[17] Sutrima; Rini Indrati, C.; Aryati, L., Stability of C0 -quasi semigroups in Banach spaces, Proceedings of the 1st Ahmad Dahlan International Conference on Mathematics and Mathematics Education, AD-INTERCOMME 2017
[18] Bárcenas, D.; Leiva, H.; Tineo Moya, A., The dual quasisemigroup and controllability of evolution equations, Journal of Mathematical Analysis and Applications, 320, 2, 691-702, (2006) · Zbl 1108.47038
[19] Megan, M.; Cuc, V., On Exponential Stability of C0-Quasi-Semigroups in Banach Spaces, Le Matematiche, 54, 2, 229-241 (2001), (1999)
[20] Chicone, C.; Latushkin, Y., Evolution Semigroups in Dynamical Systems and Differential Equations. Evolution Semigroups in Dynamical Systems and Differential Equations, Mathematical Surveys and Monographs, 70, (1999), Providence, RI, USA: American Mathematical Society, Providence, RI, USA · Zbl 0970.47027
[21] Curtain, R. F.; Zwart, H., An Introduction to In finite-Dimensional Linear Systems Theory, (1995), Springer
[22] Engel, K.-J.; Nagel, R., One-Parameter Semigroups for Linear Evolution Equations. One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194, (1999), Springer
[23] Curtain, R. F.; Pritchard, A. J., Infinite Dimensional Linear Systems. Infinite Dimensional Linear Systems, Lecture Notes in Control and Information Sciences, 8, (1978), Berlin , Germany: Springer, Berlin , Germany · Zbl 0389.93001
[24] Zabczyk, J., Mathematical Control Theory: An Introduction, (1992), Boston, Mass, USA: Birkhuser, Boston, Mass, USA · Zbl 1071.93500
[25] Randhi, A., Spectral Theory for Positive Semigroups and Applications, (2002), Lece, Italy: Salento University Publishing, Lece, Italy
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.