# American Institute of Mathematical Sciences

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 and Related Fields, 2016, 6 (3) : 447-466. doi: 10.3934/mcrf.2016011
##### 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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##### 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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