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  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. 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.  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

show all references

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. 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.  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

[1]

Güher Çamliyurt, Igor Kukavica. A local asymptotic expansion for a solution of the Stokes system. Evolution Equations & 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 & Related Models, 2009, 2 (1) : 205-214. doi: 10.3934/krm.2009.2.205

[3]

Patrick Bonckaert, P. De Maesschalck. Gevrey and analytic local models for families of vector fields. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 377-400. doi: 10.3934/dcdsb.2008.10.377

[4]

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 & Optimization, 2021  doi: 10.3934/naco.2021016

[5]

Xuanji Hou, Lei Jiao. On local rigidity of reducibility of analytic quasi-periodic cocycles on $U(n)$. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 3125-3152. doi: 10.3934/dcds.2016.36.3125

[6]

Xuanji Hou, Jiangong You. Local rigidity of reducibility of analytic quasi-periodic cocycles on $U(n)$. Discrete & Continuous Dynamical Systems, 2009, 24 (2) : 441-454. doi: 10.3934/dcds.2009.24.441

[7]

Entisar A.-L. Ali, G. Charlot. Local contact sub-Finslerian geometry for maximum norms in dimension 3. Mathematical Control & Related Fields, 2021, 11 (2) : 373-401. doi: 10.3934/mcrf.2020041

[8]

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 & Optimization, 2021, 11 (1) : 75-86. doi: 10.3934/naco.2020016

[9]

Jerry L. Bona, Zoran Grujić, Henrik Kalisch. A KdV-type Boussinesq system: From the energy level to analytic spaces. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1121-1139. doi: 10.3934/dcds.2010.26.1121

[10]

Qifan Li. Local well-posedness for the periodic Korteweg-de Vries equation in analytic Gevrey classes. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1097-1109. doi: 10.3934/cpaa.2012.11.1097

[11]

Björn Sandstede, Arnd Scheel. Relative Morse indices, Fredholm indices, and group velocities. Discrete & Continuous Dynamical Systems, 2008, 20 (1) : 139-158. doi: 10.3934/dcds.2008.20.139

[12]

Shigeaki Koike, Andrzej Świech. Local maximum principle for $L^p$-viscosity solutions of fully nonlinear elliptic PDEs with unbounded coefficients. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1897-1910. doi: 10.3934/cpaa.2012.11.1897

[13]

El Mustapha Ait Ben Hassi, Mohamed Fadili, Lahcen Maniar. Controllability of a system of degenerate parabolic equations with non-diagonalizable diffusion matrix. Mathematical Control & Related Fields, 2020, 10 (3) : 623-642. doi: 10.3934/mcrf.2020013

[14]

Rainer Buckdahn, Ingo Bulla, Jin Ma. Pathwise Taylor expansions for Itô random fields. Mathematical Control & Related Fields, 2011, 1 (4) : 437-468. doi: 10.3934/mcrf.2011.1.437

[15]

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

[16]

Marta Lewicka, Piotr B. Mucha. A local existence result for a system of viscoelasticity with physical viscosity. Evolution Equations & Control Theory, 2013, 2 (2) : 337-353. doi: 10.3934/eect.2013.2.337

[17]

Y. A. Li, P. J. Olver. Convergence of solitary-wave solutions in a perturbed bi-hamiltonian dynamical system ii. complex analytic behavior and convergence to non-analytic solutions. Discrete & Continuous Dynamical Systems, 1998, 4 (1) : 159-191. doi: 10.3934/dcds.1998.4.159

[18]

Shengzhi Zhu, Shaobo Gan, Lan Wen. Indices of singularities of robustly transitive sets. Discrete & Continuous Dynamical Systems, 2008, 21 (3) : 945-957. doi: 10.3934/dcds.2008.21.945

[19]

Muhammad Aamer Rashid, Sarfraz Ahmad, Muhammad Kamran Siddiqui, Juan L. G. Guirao, Najma Abdul Rehman. Topological indices of discrete molecular structure. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2487-2495. doi: 10.3934/dcdss.2020418

[20]

Fanwen Meng, Kiok Liang Teow, Chee Kheong Ooi, Bee Hoon Heng, Seow Yian Tay. Minimization of the coefficient of variation for patient waiting system governed by a generic maximum waiting policy. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1759-1770. doi: 10.3934/jimo.2017017

2019 Impact Factor: 0.857

Metrics

  • PDF downloads (29)
  • HTML views (42)
  • Cited by (0)

Other articles
by authors

[Back to Top]