September  2016, 6(3): 447-466. doi: 10.3934/mcrf.2016011

Characterizations of integral input-to-state stability for bilinear systems in infinite dimensions

1. 

Department of Systems Design and Informatics, Kyushu Institute of Technology, 680-4 Kawazu, Iizuka, Fukuoka 820-8502, Japan

2. 

Department of Systems Design and Informatics, Kyushu Institute of Technology, 680-4 Kawazu, Iizuka 820-8502

Received  June 2014 Revised  April 2016 Published  August 2016

For bilinear infinite-dimensional dynamical systems, we show the equivalence between uniform global asymptotic stability and integral input-to-state stability. We provide two proofs of this fact. One applies to general systems over Banach spaces. The other is restricted to Hilbert spaces, but is more constructive and results in an explicit form of iISS Lyapunov functions.
Citation: Andrii Mironchenko, Hiroshi Ito. Characterizations of integral input-to-state stability for bilinear systems in infinite dimensions. Mathematical Control & Related Fields, 2016, 6 (3) : 447-466. doi: 10.3934/mcrf.2016011
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show all references

References:
[1]

Consultants Bureau, New York, 1987. doi: 10.1007/978-1-4615-7551-1.  Google Scholar

[2]

Syst. Control Lett., 56 (2007), 87-91. doi: 10.1016/j.sysconle.2006.08.003.  Google Scholar

[3]

IEEE Transactions on Automatic Control, 45 (2000), 1082-1097. doi: 10.1109/9.863594.  Google Scholar

[4]

Oxford University Press, New York, 1998.  Google Scholar

[5]

Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6.  Google Scholar

[6]

in Proc. of the 49th IEEE Conference on Decision and Control (CDC), 2010, 6547-6552. doi: 10.1109/CDC.2010.5717779.  Google Scholar

[7]

Mathematics of Control, Signals, and Systems, 25 (2013), 1-35. doi: 10.1007/s00498-012-0090-2.  Google Scholar

[8]

SIAM Journal on Control and Optimization, 51 (2013), 1962-1987. doi: 10.1137/120881993.  Google Scholar

[9]

Mathematics of Control, Signals, and Systems, 19 (2007), 93-122. doi: 10.1007/s00498-007-0014-8.  Google Scholar

[10]

Birkhäuser, Boston, MA, 1996. doi: 10.1007/978-0-8176-4759-9.  Google Scholar

[11]

Springer-Verlag, Berlin, 1981.  Google Scholar

[12]

Journal of the Franklin Institute, 339 (2002), 211-229. doi: 10.1016/S0016-0032(02)00022-4.  Google Scholar

[13]

IEEE Transactions on Automatic Control, 55 (2010), 702-708. doi: 10.1109/TAC.2009.2037457.  Google Scholar

[14]

IEEE Transactions on Automatic Control, 58 (2013), 1192-1207. doi: 10.1109/TAC.2012.2231552.  Google Scholar

[15]

IEEE Transactions on Automatic Control, 54 (2009), 2389-2404. doi: 10.1109/TAC.2009.2028980.  Google Scholar

[16]

IEEE Transactions on Automatic Control, 51 (2006), 1626-1643. doi: 10.1109/TAC.2006.882930.  Google Scholar

[17]

Springer, Basel, 2012. doi: 10.1007/978-3-0348-0399-1.  Google Scholar

[18]

Communications in Information and Systems, 8 (2008), 413-444. doi: 10.4310/CIS.2008.v8.n4.a4.  Google Scholar

[19]

Automatica, 32 (1996), 1211-1215. doi: 10.1016/0005-1098(96)00051-9.  Google Scholar

[20]

Mathematics of Control, Signals, and Systems, 7 (1994), 95-120. doi: 10.1007/BF01211469.  Google Scholar

[21]

Automatica, 37 (2001), 857-869. doi: 10.1016/S0005-1098(01)00028-0.  Google Scholar

[22]

in 7th World Congress on Intelligent Control and Automation (WCICA), 2008, 1188-1193. Google Scholar

[23]

IMA Journal of Mathematical Control and Information, 28 (2011), 309-344. doi: 10.1093/imamci/dnr001.  Google Scholar

[24]

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[25]

Automatica, 37 (2001), 637-662. doi: 10.1016/S0005-1098(01)00002-4.  Google Scholar

[26]

SIAM Journal on Control and Optimization, 34 (1996), 124-160. doi: 10.1137/S0363012993259981.  Google Scholar

[27]

SIAM Journal on Control and Optimization, 51 (2013), 1203-1231. doi: 10.1137/110850396.  Google Scholar

[28]

Mathematical Control and Related Fields, 1 (2011), 231-250. doi: 10.3934/mcrf.2011.1.231.  Google Scholar

[29]

Systems & Control Letters, 87 (2016), 23-28. doi: 10.1016/j.sysconle.2015.10.014.  Google Scholar

[30]

in Proc. of the 53th IEEE Conference on Decision and Control, 2014, 3155-3160. doi: 10.1109/CDC.2014.7039876.  Google Scholar

[31]

SIAM Journal on Control and Optimization, 53 (2015), 3364-3382. doi: 10.1137/14097269X.  Google Scholar

[32]

Kluwer Academic Publishers Group, Dordrecht, 1991. doi: 10.1007/978-94-011-3562-7.  Google Scholar

[33]

Systems & Control Letters, 55 (2006), 1006-1014. doi: 10.1016/j.sysconle.2006.06.013.  Google Scholar

[34]

Mathematics of Control, Signals, and Systems, 24 (2012), 111-134. doi: 10.1007/s00498-012-0074-2.  Google Scholar

[35]

Prentice Hall, 2010. Google Scholar

[36]

IEEE Transactions on Automatic Control, 40 (1995), 1476-1478. doi: 10.1109/9.402246.  Google Scholar

[37]

IEEE Transactions on Automatic Control, 34 (1989), 435-443. doi: 10.1109/9.28018.  Google Scholar

[38]

Systems & Control Letters, 24 (1995), 351-359. doi: 10.1016/0167-6911(94)00050-6.  Google Scholar

[39]

Systems & Control Letters, 34 (1998), 93-100. doi: 10.1016/S0167-6911(98)00003-6.  Google Scholar

[40]

in Nonlinear and Optimal Control Theory, Springer, Heidelberg, 1932 (2008), 163-220. doi: 10.1007/978-3-540-77653-6_3.  Google Scholar

[41]

American Mathematical Society, 2012. doi: 10.1090/gsm/140.  Google Scholar

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