# American Institute of Mathematical Sciences

doi: 10.3934/mcrf.2021029
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## Local Kalman rank condition for linear time varying systems

* Corresponding author: Hamid Maarouf

Received  August 2020 Revised  January 2021 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 & Related Fields, doi: 10.3934/mcrf.2021029
##### References:
 [1] A. Chang, An algebraic characterization of controllability, IEEE Transactions on Automatic Control, 10 (1965), 112-113.   Google Scholar [2] J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, vol. 136, AMS, Providence, RI, 2007. doi: 10.1090/surv/136.  Google Scholar [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.  Google Scholar [4] F. A. Khodja, A. Benabdallah, C. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] H. H. Rosenbrock, State-Space and Multivariable Theory, John Wiley & Sons, Inc., New York, 1970.  Google Scholar [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.  Google Scholar

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##### References:
 [1] A. Chang, An algebraic characterization of controllability, IEEE Transactions on Automatic Control, 10 (1965), 112-113.   Google Scholar [2] J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, vol. 136, AMS, Providence, RI, 2007. doi: 10.1090/surv/136.  Google Scholar [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.  Google Scholar [4] F. A. Khodja, A. Benabdallah, C. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] H. H. Rosenbrock, State-Space and Multivariable Theory, John Wiley & Sons, Inc., New York, 1970.  Google Scholar [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.  Google Scholar
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