June  2022, 12(2): 433-446. doi: 10.3934/mcrf.2021029

Local Kalman rank condition for linear time varying systems

High School of Technology of Safi, Cadi Ayyad University, Safi, Morocco

* Corresponding author: Hamid Maarouf

Received  August 2020 Revised  January 2021 Published  June 2022 Early access  May 2021

In this paper, we study some non-negative integers related to a linear time varying system and to some Krylov sub-spaces associated to this system. Such integers are similar to the controllability indices and have been used in the literature to derive results on the controllability of linear systems. The purpose of this paper goes in the same direction by studying the local behavior of these integers especially nearby instants in the time interval with some maximal rank condition and then apply them to get some results which generalize the mentioned existing results.

Citation: Hamid Maarouf. Local Kalman rank condition for linear time varying systems. Mathematical Control and Related Fields, 2022, 12 (2) : 433-446. doi: 10.3934/mcrf.2021029
References:
[1]

A. Chang, An algebraic characterization of controllability, IEEE Transactions on Automatic Control, 10 (1965), 112-113. 

[2]

J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, vol. 136, AMS, Providence, RI, 2007. doi: 10.1090/surv/136.

[3]

G. Dauphin-Tanguy and C. Sueur, Controllability indices for structured systems, Linear Algebra and its Applications, 250 (1997), 275-287.  doi: 10.1016/0024-3795(95)00598-6.

[4]

F. A. KhodjaA. BenabdallahC. Dupaix and M. Gonzàlez-Burgos, A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems, Differential Equations and Applications, 1 (2009), 427-457.  doi: 10.7153/dea-01-24.

[5]

H. Maarouf, Controllability and nonsingular solutions of Sylvester equations, Electronic Journal of Linear Algebra, 31 (2016), 721-739.  doi: 10.13001/1081-3810.3106.

[6]

H. Maarouf, Controllable subspace for linear time varying systems, European Journal of Control, 50 (2019), 72-78.  doi: 10.1016/j.ejcon.2019.05.001.

[7]

H. H. Rosenbrock, State-Space and Multivariable Theory, John Wiley & Sons, Inc., New York, 1970.

[8]

L. M. Silverman and H. E. Meadows, Controllability and observability in time-variable linear systems, SIAM Journal on Control, 5 (1967), 64-73.  doi: 10.1137/0305005.

show all references

References:
[1]

A. Chang, An algebraic characterization of controllability, IEEE Transactions on Automatic Control, 10 (1965), 112-113. 

[2]

J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, vol. 136, AMS, Providence, RI, 2007. doi: 10.1090/surv/136.

[3]

G. Dauphin-Tanguy and C. Sueur, Controllability indices for structured systems, Linear Algebra and its Applications, 250 (1997), 275-287.  doi: 10.1016/0024-3795(95)00598-6.

[4]

F. A. KhodjaA. BenabdallahC. Dupaix and M. Gonzàlez-Burgos, A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems, Differential Equations and Applications, 1 (2009), 427-457.  doi: 10.7153/dea-01-24.

[5]

H. Maarouf, Controllability and nonsingular solutions of Sylvester equations, Electronic Journal of Linear Algebra, 31 (2016), 721-739.  doi: 10.13001/1081-3810.3106.

[6]

H. Maarouf, Controllable subspace for linear time varying systems, European Journal of Control, 50 (2019), 72-78.  doi: 10.1016/j.ejcon.2019.05.001.

[7]

H. H. Rosenbrock, State-Space and Multivariable Theory, John Wiley & Sons, Inc., New York, 1970.

[8]

L. M. Silverman and H. E. Meadows, Controllability and observability in time-variable linear systems, SIAM Journal on Control, 5 (1967), 64-73.  doi: 10.1137/0305005.

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