# American Institute of Mathematical Sciences

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

## Local Kalman rank condition for linear time varying systems

* 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. 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. [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. 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. [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.
 [1] Güher Çamliyurt, Igor Kukavica. A local asymptotic expansion for a solution of the Stokes system. Evolution Equations and Control Theory, 2016, 5 (4) : 647-659. doi: 10.3934/eect.2016023 [2] Yan Guo, Juhi Jang, Ning Jiang. Local Hilbert expansion for the Boltzmann equation. Kinetic and Related Models, 2009, 2 (1) : 205-214. doi: 10.3934/krm.2009.2.205 [3] Quan Zhou, Yabing Sun. High order one-step methods for backward stochastic differential equations via Itô-Taylor expansion. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021233 [4] Patrick Bonckaert, P. De Maesschalck. Gevrey and analytic local models for families of vector fields. Discrete and Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 377-400. doi: 10.3934/dcdsb.2008.10.377 [5] Xin Du, M. Monir Uddin, A. Mostakim Fony, Md. Tanzim Hossain, Md. Nazmul Islam Shuzan. Iterative Rational Krylov Algorithms for model reduction of a class of constrained structural dynamic system with Engineering applications. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021016 [6] Xuanji Hou, Lei Jiao. On local rigidity of reducibility of analytic quasi-periodic cocycles on $U(n)$. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3125-3152. doi: 10.3934/dcds.2016.36.3125 [7] Xuanji Hou, Jiangong You. Local rigidity of reducibility of analytic quasi-periodic cocycles on $U(n)$. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 441-454. doi: 10.3934/dcds.2009.24.441 [8] Entisar A.-L. Ali, G. Charlot. Local contact sub-Finslerian geometry for maximum norms in dimension 3. Mathematical Control and Related Fields, 2021, 11 (2) : 373-401. doi: 10.3934/mcrf.2020041 [9] Jian Song, Meng Wang. Stochastic maximum principle for systems driven by local martingales with spatial parameters. Probability, Uncertainty and Quantitative Risk, 2021, 6 (3) : 213-236. doi: 10.3934/puqr.2021011 [10] Sihem Guerarra. Maximum and minimum ranks and inertias of the Hermitian parts of the least rank solution of the matrix equation AXB = C. Numerical Algebra, Control and Optimization, 2021, 11 (1) : 75-86. doi: 10.3934/naco.2020016 [11] Jerry L. Bona, Zoran Grujić, Henrik Kalisch. A KdV-type Boussinesq system: From the energy level to analytic spaces. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1121-1139. doi: 10.3934/dcds.2010.26.1121 [12] Björn Sandstede, Arnd Scheel. Relative Morse indices, Fredholm indices, and group velocities. Discrete and Continuous Dynamical Systems, 2008, 20 (1) : 139-158. doi: 10.3934/dcds.2008.20.139 [13] Qifan Li. Local well-posedness for the periodic Korteweg-de Vries equation in analytic Gevrey classes. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1097-1109. doi: 10.3934/cpaa.2012.11.1097 [14] Shigeaki Koike, Andrzej Świech. Local maximum principle for $L^p$-viscosity solutions of fully nonlinear elliptic PDEs with unbounded coefficients. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1897-1910. doi: 10.3934/cpaa.2012.11.1897 [15] El Mustapha Ait Ben Hassi, Mohamed Fadili, Lahcen Maniar. Controllability of a system of degenerate parabolic equations with non-diagonalizable diffusion matrix. Mathematical Control and Related Fields, 2020, 10 (3) : 623-642. doi: 10.3934/mcrf.2020013 [16] Kazuhiro Kurata, Yuki Osada. Asymptotic expansion of the ground state energy for nonlinear Schrödinger system with three wave interaction. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4239-4251. doi: 10.3934/cpaa.2021157 [17] Rainer Buckdahn, Ingo Bulla, Jin Ma. Pathwise Taylor expansions for Itô random fields. Mathematical Control and Related Fields, 2011, 1 (4) : 437-468. doi: 10.3934/mcrf.2011.1.437 [18] Uchida Hidetake. Analytic smoothing effect and global existence of small solutions for the elliptic-hyperbolic Davey-Stewartson system. Conference Publications, 2001, 2001 (Special) : 182-190. doi: 10.3934/proc.2001.2001.182 [19] Shengzhi Zhu, Shaobo Gan, Lan Wen. Indices of singularities of robustly transitive sets. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 945-957. doi: 10.3934/dcds.2008.21.945 [20] Muhammad Aamer Rashid, Sarfraz Ahmad, Muhammad Kamran Siddiqui, Juan L. G. Guirao, Najma Abdul Rehman. Topological indices of discrete molecular structure. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2487-2495. doi: 10.3934/dcdss.2020418

2020 Impact Factor: 1.284