×

Stabilizability and controllability for systems of difference equations. (English) Zbl 1108.93055

Summary: We establish the connection between exact controllability and complete stabilizability for systems of difference equations. We prove that the complete stabilizability of a system of difference equations \((A,B)\) with \(A\) surjective implies the exact controllability. By examples we motivate the condition on the surjectivity of \(A\) and we show that generally, the converse implication is not valid.

MSC:

93C55 Discrete-time control/observation systems
93D15 Stabilization of systems by feedback
93B05 Controllability
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1137/S0363012996313835 · Zbl 0935.93037 · doi:10.1137/S0363012996313835
[2] DOI: 10.1007/BF02551372 · Zbl 0715.93038 · doi:10.1007/BF02551372
[3] Barbu V., Analysis and Control of Nonlinear Infinite Dimensional Systems (1993) · Zbl 0776.49005
[4] Barbu V., Mathematical Methods in Optimization of Differential Systems (1994) · Zbl 0819.49002
[5] DOI: 10.1080/10236199908808210 · Zbl 0954.39011 · doi:10.1080/10236199908808210
[6] Elaydi S., An Introduction to Difference Equations, Undergraduate Texts in Mathematics (2005) · Zbl 1071.39001
[7] DOI: 10.1023/B:DIEQ.0000046864.27700.e9 · Zbl 1160.93309 · doi:10.1023/B:DIEQ.0000046864.27700.e9
[8] DOI: 10.1016/S0167-6911(03)00116-6 · Zbl 1157.93468 · doi:10.1016/S0167-6911(03)00116-6
[9] Hinrichsen D., Mathematical Systems Theory I. Modelling, State Space Analysis, Stability and Robustness, Texts in Applied Mathematics 48 (2005) · Zbl 1074.93003
[10] Klamka J., International Journal of Applied Mathematics and Computer Science 12 pp 173– (2002)
[11] Kocic V.L., Global Behavior of Nonlinear Difference Equations of Higher Order (1993) · Zbl 0787.39001 · doi:10.1007/978-94-017-1703-8
[12] Komornik V., Exact Controllability and Stabilization–The Multiplier Method (1994) · Zbl 0937.93003
[13] Komornik V., Control Cybernet 28 pp 813– (1999)
[14] DOI: 10.1080/1023619031000092184 · Zbl 1064.93006 · doi:10.1080/1023619031000092184
[15] Lions J.-L., Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués, Tome 1 (1988)
[16] Megan M., Seminarul de Ecuţii Funcţionale, Universitatea din Timişoara 32 pp 1– (1975)
[17] Megan M., Revista Matematica Complutense 15 pp 599– (2002)
[18] Megan M., Journal of Discrete and Continuous Dynamical Systems 9 pp 383– (2003)
[19] DOI: 10.1080/00207178808906333 · Zbl 0666.93102 · doi:10.1080/00207178808906333
[20] DOI: 10.1080/10236190412331314178 · Zbl 1064.39011 · doi:10.1080/10236190412331314178
[21] Sasu A.L., Applied Mathematics Letters (2006)
[22] DOI: 10.1016/j.jmaa.2005.04.047 · Zbl 1098.39014 · doi:10.1016/j.jmaa.2005.04.047
[23] DOI: 10.1023/B:AURC.0000023533.13882.13 · Zbl 1095.93025 · doi:10.1023/B:AURC.0000023533.13882.13
[24] DOI: 10.1093/imamci/11.3.253 · Zbl 0812.93055 · doi:10.1093/imamci/11.3.253
[25] Ya Z., Journal of Difference Equations and Applications 3 pp 539– (1998)
[26] Zabczyk J., Seminarul de Ecuaţii Funcţionale, Universitatea din Timişoara 38 pp 1– (1976)
[27] Zabczyk J., Mathematical Control Theory: An Introduction (1995) · Zbl 1071.93500
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.