# zbMATH — the first resource for mathematics

Stabilizability of linear time-varying systems. (English) Zbl 1280.93068
Summary: For linear time-varying systems with bounded system matrices we discuss the problem of stabilizability by linear state feedback. For example, it is shown that complete controllability implies the existence of a feedback so that the closed-loop system is asymptotically stable. We also show that the system is completely controllable if, and only if, the Lyapunov exponent is arbitrarily assignable by a suitable feedback. For uniform exponential stabilizability and the assignability of the Bohl exponent this property is known. Also, dynamic feedback does not provide more freedom to address the stabilization problem. The unifying tools for our results are two finite $$(L^2)$$ cost conditions. The distinction of exponential and uniform exponential stabilizability is then a question of whether the finite cost condition is uniform in the initial time or not.

##### MSC:
 93D15 Stabilization of systems by feedback 49N10 Linear-quadratic optimal control problems 93C05 Linear systems in control theory 93D20 Asymptotic stability in control theory
Full Text:
##### References:
 [1] Ikeda, M.; Maeda, H.; Kodama, S., Stabilization of linear systems, SIAM J. Control Optim., 10, 716-729, (1972) · Zbl 0244.93049 [2] Ikeda, M.; Maeda, H.; Kodama, S., Estimation and feedback in linear time-varying systems: a deterministic theory, SIAM J. Control Optim., 13, 304-326, (1975) · Zbl 0305.93037 [3] Anderson, B. D.O.; Moore, J. B., Linear optimal control, (1971), Prentice-Hall Englewood Cliffs, NJ · Zbl 0321.49001 [4] Ravi, R.; Pascoal, A. M.; Khargonekar, P. P., Normalized coprime factorizations for linear time-varying systems, Systems Control Lett., 18, 6, 455-465, (1992) · Zbl 0765.93045 [5] Rotea, M. A.; Khargonekar, P. P., Stabilizability of linear time-varying and uncertain linear systems, IEEE Trans. Automat. Control, 33, 9, 884-887, (1988) · Zbl 0646.93051 [6] Phat, V. N.; Ha, Q., New characterization of controllability via stabilizability and Riccati equation for LTV systems, IMA J. Math. Control Inform., 25, 4, 419-429, (2008) · Zbl 1152.93030 [7] Hinrichsen, D.; Pritchard, A. J., (Mathematical Systems Theory I. Modelling, State Space Analysis, Stability and Robustness, Texts in Applied Mathematics, vol. 48, (2005), Springer-Verlag Berlin) · Zbl 1074.93003 [8] Daleckiĭ, J. L.; Kreĭn, M. G., (Stability of Solutions of Differential Equations in Banach Spaces, Translations of Mathematical Monographs, vol. 43, (1974), American Mathematical Society Providence, RI) [9] Kalman, R. E., Contributions to the theory of optimal control, Bol. Soc. Mat. Mexico Ser. II, 5, 102-119, (1960) [10] Cheng, A., A direct way to stabilize continuous-time and discrete-time linear time-varying systems, IEEE Trans. Automat. Control, AC-24, 4, 641-643, (1979) · Zbl 0412.93049 [11] Rugh, W. J., (Linear System Theory, Information and System Sciences Series, (1996), Prentice-Hall NJ) [12] Wonham, W. M., On pole assignment in multi-input controllable linear systems, IEEE Trans. Automat. Control, AC-12, 660-665, (1967) [13] Sontag, E. D., Mathematical control theory: deterministic finite dimensional systems, (1998), Springer-Verlag New York · Zbl 0945.93001 [14] Weiss, G.; Rebarber, R., Optimizability and estimatability for infinite-dimensional linear systems, SIAM J. Control Optim., 39, 4, 1204-1232, (2000) · Zbl 0981.93032 [15] Knobloch, H. W.; Kwakernaak, H., Lineare kontrolltheorie, (1985), Springer-Verlag Berlin · Zbl 0574.93001 [16] Anderson, B. D.O.; Moore, J. B., Detectability and stabilizability of time-varying discrete-time linear systems, SIAM J. Control Optim., 19, 1, 20-32, (1981) · Zbl 0468.93051
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.