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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 |
References:
[1] |
V. M. Alekseev, V. M. Tikhomirov and S. V. Fomin, Optimal Control, Consultants Bureau, New York, 1987.
doi: 10.1007/978-1-4615-7551-1. |
[2] |
D. Angeli and A. Astolfi, A tight small gain theorem for not necessarily ISS systems, Syst. Control Lett., 56 (2007), 87-91.
doi: 10.1016/j.sysconle.2006.08.003. |
[3] |
D. Angeli, E. D. Sontag and Y. Wang, A characterization of integral input-to-state stability, IEEE Transactions on Automatic Control, 45 (2000), 1082-1097.
doi: 10.1109/9.863594. |
[4] |
T. Cazenave and A. Haraux, An Introduction To Semilinear Evolution Equations, Oxford University Press, New York, 1998. |
[5] |
R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Springer-Verlag, New York, 1995.
doi: 10.1007/978-1-4612-4224-6. |
[6] |
S. Dashkovskiy and A. Mironchenko, On the uniform input-to-state stability of reaction-diffusion systems, in Proc. of the 49th IEEE Conference on Decision and Control (CDC), 2010, 6547-6552.
doi: 10.1109/CDC.2010.5717779. |
[7] |
S. Dashkovskiy and A. Mironchenko, Input-to-state stability of infinite-dimensional control systems, Mathematics of Control, Signals, and Systems, 25 (2013), 1-35.
doi: 10.1007/s00498-012-0090-2. |
[8] |
S. Dashkovskiy and A. Mironchenko, Input-to-state stability of nonlinear impulsive systems, SIAM Journal on Control and Optimization, 51 (2013), 1962-1987.
doi: 10.1137/120881993. |
[9] |
S. Dashkovskiy, B. S. Rüffer and F. R. Wirth, An ISS small gain theorem for general networks, Mathematics of Control, Signals, and Systems, 19 (2007), 93-122.
doi: 10.1007/s00498-007-0014-8. |
[10] |
R. A. Freeman and P. V. Kokotovic, Robust Nonlinear Control Design: State-Space and Lyapunov Techniques, Birkhäuser, Boston, MA, 1996.
doi: 10.1007/978-0-8176-4759-9. |
[11] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, 1981. |
[12] |
B. Ingalls and E. D. Sontag, A small-gain theorem with applications to input/output systems, incremental stability, detectability, and interconnections, Journal of the Franklin Institute, 339 (2002), 211-229.
doi: 10.1016/S0016-0032(02)00022-4. |
[13] |
H. Ito, A Lyapunov Approach to Cascade Interconnection of Integral Input-to-State Stable Systems, IEEE Transactions on Automatic Control, 55 (2010), 702-708.
doi: 10.1109/TAC.2009.2037457. |
[14] |
H. Ito, Z. P. Jiang, S. Dashkovskiy and B. Rüffer, Robust stability of networks of iISS systems: Construction of sum-type Lyapunov functions, IEEE Transactions on Automatic Control, 58 (2013), 1192-1207.
doi: 10.1109/TAC.2012.2231552. |
[15] |
H. Ito and Z.-P. Jiang, Necessary and Sufficient Small Gain Conditions for Integral Input-to-State Stable Systems: A Lyapunov Perspective, IEEE Transactions on Automatic Control, 54 (2009), 2389-2404.
doi: 10.1109/TAC.2009.2028980. |
[16] |
H. Ito, State-dependent scaling problems and stability of interconnected iISS and ISS systems, IEEE Transactions on Automatic Control, 51 (2006), 1626-1643.
doi: 10.1109/TAC.2006.882930. |
[17] |
B. Jacob and H. J. Zwart, Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces, Springer, Basel, 2012.
doi: 10.1007/978-3-0348-0399-1. |
[18] |
B. Jayawardhana, H. Logemann and E. P. Ryan, Infinite-dimensional feedback systems: the circle criterion and input-to-state stability, Communications in Information and Systems, 8 (2008), 413-444.
doi: 10.4310/CIS.2008.v8.n4.a4. |
[19] |
Z.-P. Jiang, I. M. Y. Mareels and Y. Wang, A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems, Automatica, 32 (1996), 1211-1215.
doi: 10.1016/0005-1098(96)00051-9. |
[20] |
Z.-P. Jiang, A. R. Teel and L. Praly, Small-gain theorem for ISS systems and applications, Mathematics of Control, Signals, and Systems, 7 (1994), 95-120.
doi: 10.1007/BF01211469. |
[21] |
Z.-P. Jiang and Y. Wang, Input-to-state stability for discrete-time nonlinear systems, Automatica, 37 (2001), 857-869.
doi: 10.1016/S0005-1098(01)00028-0. |
[22] |
Z.-P. Jiang and Y. Wang, A generalization of the nonlinear small-gain theorem for large-scale complex systems, in 7th World Congress on Intelligent Control and Automation (WCICA), 2008, 1188-1193. |
[23] |
I. Karafyllis and Z.-P. Jiang, A vector small-gain theorem for general non-linear control systems, IMA Journal of Mathematical Control and Information, 28 (2011), 309-344.
doi: 10.1093/imamci/dnr001. |
[24] |
I. Karafyllis and Z.-P. Jiang, A new small-gain theorem with an application to the stabilization of the chemostat, Int. J. Robust. Nonlinear Control, 22 (2012), 1602-1630.
doi: 10.1002/rnc.1773. |
[25] |
P. Kokotović and M. Arcak, Constructive nonlinear control: a historical perspective, Automatica, 37 (2001), 637-662.
doi: 10.1016/S0005-1098(01)00002-4. |
[26] |
Y. Lin, E. D. Sontag and Y. Wang, A smooth converse Lyapunov theorem for robust stability, SIAM Journal on Control and Optimization, 34 (1996), 124-160.
doi: 10.1137/S0363012993259981. |
[27] |
H. Logemann, Stabilization of well-posed infinite-dimensional systems by dynamic sampled-data feedback, SIAM Journal on Control and Optimization, 51 (2013), 1203-1231.
doi: 10.1137/110850396. |
[28] |
F. Mazenc and C. Prieur, Strict Lyapunov functions for semilinear parabolic partial differential equations, Mathematical Control and Related Fields, 1 (2011), 231-250.
doi: 10.3934/mcrf.2011.1.231. |
[29] |
A. Mironchenko, Local input-to-state stability: Characterizations and counterexamples, Systems & Control Letters, 87 (2016), 23-28.
doi: 10.1016/j.sysconle.2015.10.014. |
[30] |
A. Mironchenko and H. Ito, Integral input-to-state stability of bilinear infinite-dimensional systems, in Proc. of the 53th IEEE Conference on Decision and Control, 2014, 3155-3160.
doi: 10.1109/CDC.2014.7039876. |
[31] |
A. Mironchenko and H. Ito, Construction of Lyapunov functions for interconnected parabolic systems: An iISS approach, SIAM Journal on Control and Optimization, 53 (2015), 3364-3382.
doi: 10.1137/14097269X. |
[32] |
D. S. Mitrinović, J. E. Pečarić and A. M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives, Kluwer Academic Publishers Group, Dordrecht, 1991.
doi: 10.1007/978-94-011-3562-7. |
[33] |
P. Pepe and Z.-P. Jiang, A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems, Systems & Control Letters, 55 (2006), 1006-1014.
doi: 10.1016/j.sysconle.2006.06.013. |
[34] |
C. Prieur and F. Mazenc, ISS-Lyapunov functions for time-varying hyperbolic systems of balance laws, Mathematics of Control, Signals, and Systems, 24 (2012), 111-134.
doi: 10.1007/s00498-012-0074-2. |
[35] |
H. Royden and P. Fitzpatrick, Real Analysis, Prentice Hall, 2010. |
[36] |
E. Sontag and A. Teel, Changing supply functions in input/state stable systems, IEEE Transactions on Automatic Control, 40 (1995), 1476-1478.
doi: 10.1109/9.402246. |
[37] |
E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Transactions on Automatic Control, 34 (1989), 435-443.
doi: 10.1109/9.28018. |
[38] |
E. D. Sontag and Y. Wang, On characterizations of the input-to-state stability property, Systems & Control Letters, 24 (1995), 351-359.
doi: 10.1016/0167-6911(94)00050-6. |
[39] |
E. D. Sontag, Comments on integral variants of ISS, Systems & Control Letters, 34 (1998), 93-100.
doi: 10.1016/S0167-6911(98)00003-6. |
[40] |
E. D. Sontag, Input to state stability: Basic concepts and results, in Nonlinear and Optimal Control Theory, Springer, Heidelberg, 1932 (2008), 163-220.
doi: 10.1007/978-3-540-77653-6_3. |
[41] |
G. Teschl, Ordinary Differential Equations and Dynamical Systems, American Mathematical Society, 2012.
doi: 10.1090/gsm/140. |
show all references
References:
[1] |
V. M. Alekseev, V. M. Tikhomirov and S. V. Fomin, Optimal Control, Consultants Bureau, New York, 1987.
doi: 10.1007/978-1-4615-7551-1. |
[2] |
D. Angeli and A. Astolfi, A tight small gain theorem for not necessarily ISS systems, Syst. Control Lett., 56 (2007), 87-91.
doi: 10.1016/j.sysconle.2006.08.003. |
[3] |
D. Angeli, E. D. Sontag and Y. Wang, A characterization of integral input-to-state stability, IEEE Transactions on Automatic Control, 45 (2000), 1082-1097.
doi: 10.1109/9.863594. |
[4] |
T. Cazenave and A. Haraux, An Introduction To Semilinear Evolution Equations, Oxford University Press, New York, 1998. |
[5] |
R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Springer-Verlag, New York, 1995.
doi: 10.1007/978-1-4612-4224-6. |
[6] |
S. Dashkovskiy and A. Mironchenko, On the uniform input-to-state stability of reaction-diffusion systems, in Proc. of the 49th IEEE Conference on Decision and Control (CDC), 2010, 6547-6552.
doi: 10.1109/CDC.2010.5717779. |
[7] |
S. Dashkovskiy and A. Mironchenko, Input-to-state stability of infinite-dimensional control systems, Mathematics of Control, Signals, and Systems, 25 (2013), 1-35.
doi: 10.1007/s00498-012-0090-2. |
[8] |
S. Dashkovskiy and A. Mironchenko, Input-to-state stability of nonlinear impulsive systems, SIAM Journal on Control and Optimization, 51 (2013), 1962-1987.
doi: 10.1137/120881993. |
[9] |
S. Dashkovskiy, B. S. Rüffer and F. R. Wirth, An ISS small gain theorem for general networks, Mathematics of Control, Signals, and Systems, 19 (2007), 93-122.
doi: 10.1007/s00498-007-0014-8. |
[10] |
R. A. Freeman and P. V. Kokotovic, Robust Nonlinear Control Design: State-Space and Lyapunov Techniques, Birkhäuser, Boston, MA, 1996.
doi: 10.1007/978-0-8176-4759-9. |
[11] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, 1981. |
[12] |
B. Ingalls and E. D. Sontag, A small-gain theorem with applications to input/output systems, incremental stability, detectability, and interconnections, Journal of the Franklin Institute, 339 (2002), 211-229.
doi: 10.1016/S0016-0032(02)00022-4. |
[13] |
H. Ito, A Lyapunov Approach to Cascade Interconnection of Integral Input-to-State Stable Systems, IEEE Transactions on Automatic Control, 55 (2010), 702-708.
doi: 10.1109/TAC.2009.2037457. |
[14] |
H. Ito, Z. P. Jiang, S. Dashkovskiy and B. Rüffer, Robust stability of networks of iISS systems: Construction of sum-type Lyapunov functions, IEEE Transactions on Automatic Control, 58 (2013), 1192-1207.
doi: 10.1109/TAC.2012.2231552. |
[15] |
H. Ito and Z.-P. Jiang, Necessary and Sufficient Small Gain Conditions for Integral Input-to-State Stable Systems: A Lyapunov Perspective, IEEE Transactions on Automatic Control, 54 (2009), 2389-2404.
doi: 10.1109/TAC.2009.2028980. |
[16] |
H. Ito, State-dependent scaling problems and stability of interconnected iISS and ISS systems, IEEE Transactions on Automatic Control, 51 (2006), 1626-1643.
doi: 10.1109/TAC.2006.882930. |
[17] |
B. Jacob and H. J. Zwart, Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces, Springer, Basel, 2012.
doi: 10.1007/978-3-0348-0399-1. |
[18] |
B. Jayawardhana, H. Logemann and E. P. Ryan, Infinite-dimensional feedback systems: the circle criterion and input-to-state stability, Communications in Information and Systems, 8 (2008), 413-444.
doi: 10.4310/CIS.2008.v8.n4.a4. |
[19] |
Z.-P. Jiang, I. M. Y. Mareels and Y. Wang, A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems, Automatica, 32 (1996), 1211-1215.
doi: 10.1016/0005-1098(96)00051-9. |
[20] |
Z.-P. Jiang, A. R. Teel and L. Praly, Small-gain theorem for ISS systems and applications, Mathematics of Control, Signals, and Systems, 7 (1994), 95-120.
doi: 10.1007/BF01211469. |
[21] |
Z.-P. Jiang and Y. Wang, Input-to-state stability for discrete-time nonlinear systems, Automatica, 37 (2001), 857-869.
doi: 10.1016/S0005-1098(01)00028-0. |
[22] |
Z.-P. Jiang and Y. Wang, A generalization of the nonlinear small-gain theorem for large-scale complex systems, in 7th World Congress on Intelligent Control and Automation (WCICA), 2008, 1188-1193. |
[23] |
I. Karafyllis and Z.-P. Jiang, A vector small-gain theorem for general non-linear control systems, IMA Journal of Mathematical Control and Information, 28 (2011), 309-344.
doi: 10.1093/imamci/dnr001. |
[24] |
I. Karafyllis and Z.-P. Jiang, A new small-gain theorem with an application to the stabilization of the chemostat, Int. J. Robust. Nonlinear Control, 22 (2012), 1602-1630.
doi: 10.1002/rnc.1773. |
[25] |
P. Kokotović and M. Arcak, Constructive nonlinear control: a historical perspective, Automatica, 37 (2001), 637-662.
doi: 10.1016/S0005-1098(01)00002-4. |
[26] |
Y. Lin, E. D. Sontag and Y. Wang, A smooth converse Lyapunov theorem for robust stability, SIAM Journal on Control and Optimization, 34 (1996), 124-160.
doi: 10.1137/S0363012993259981. |
[27] |
H. Logemann, Stabilization of well-posed infinite-dimensional systems by dynamic sampled-data feedback, SIAM Journal on Control and Optimization, 51 (2013), 1203-1231.
doi: 10.1137/110850396. |
[28] |
F. Mazenc and C. Prieur, Strict Lyapunov functions for semilinear parabolic partial differential equations, Mathematical Control and Related Fields, 1 (2011), 231-250.
doi: 10.3934/mcrf.2011.1.231. |
[29] |
A. Mironchenko, Local input-to-state stability: Characterizations and counterexamples, Systems & Control Letters, 87 (2016), 23-28.
doi: 10.1016/j.sysconle.2015.10.014. |
[30] |
A. Mironchenko and H. Ito, Integral input-to-state stability of bilinear infinite-dimensional systems, in Proc. of the 53th IEEE Conference on Decision and Control, 2014, 3155-3160.
doi: 10.1109/CDC.2014.7039876. |
[31] |
A. Mironchenko and H. Ito, Construction of Lyapunov functions for interconnected parabolic systems: An iISS approach, SIAM Journal on Control and Optimization, 53 (2015), 3364-3382.
doi: 10.1137/14097269X. |
[32] |
D. S. Mitrinović, J. E. Pečarić and A. M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives, Kluwer Academic Publishers Group, Dordrecht, 1991.
doi: 10.1007/978-94-011-3562-7. |
[33] |
P. Pepe and Z.-P. Jiang, A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems, Systems & Control Letters, 55 (2006), 1006-1014.
doi: 10.1016/j.sysconle.2006.06.013. |
[34] |
C. Prieur and F. Mazenc, ISS-Lyapunov functions for time-varying hyperbolic systems of balance laws, Mathematics of Control, Signals, and Systems, 24 (2012), 111-134.
doi: 10.1007/s00498-012-0074-2. |
[35] |
H. Royden and P. Fitzpatrick, Real Analysis, Prentice Hall, 2010. |
[36] |
E. Sontag and A. Teel, Changing supply functions in input/state stable systems, IEEE Transactions on Automatic Control, 40 (1995), 1476-1478.
doi: 10.1109/9.402246. |
[37] |
E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Transactions on Automatic Control, 34 (1989), 435-443.
doi: 10.1109/9.28018. |
[38] |
E. D. Sontag and Y. Wang, On characterizations of the input-to-state stability property, Systems & Control Letters, 24 (1995), 351-359.
doi: 10.1016/0167-6911(94)00050-6. |
[39] |
E. D. Sontag, Comments on integral variants of ISS, Systems & Control Letters, 34 (1998), 93-100.
doi: 10.1016/S0167-6911(98)00003-6. |
[40] |
E. D. Sontag, Input to state stability: Basic concepts and results, in Nonlinear and Optimal Control Theory, Springer, Heidelberg, 1932 (2008), 163-220.
doi: 10.1007/978-3-540-77653-6_3. |
[41] |
G. Teschl, Ordinary Differential Equations and Dynamical Systems, American Mathematical Society, 2012.
doi: 10.1090/gsm/140. |
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