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Input-to-state stability of infinite-dimensional stochastic nonlinear systems
Department of Mathematics, Harbin Institute of Technology at Weihai, Weihai 264209, PR China |
In this paper, the input-to-state stability (ISS), stochastic-ISS (SISS) and integral-ISS (iISS) for mild solutions of infinite-dimensional stochastic nonlinear systems (IDSNS) are investigated, respectively. By constructing a class of Yosida strong solution approximating systems for IDSNS and using the infinite-dimensional version Itô's formula, Lyapunov-based sufficient criteria are derived for ensuring ISS-type properties of IDSNS, which extend the existing corresponding results of infinite-dimensional deterministic systems. Moreover, two examples are presented to demonstrate the main results.
References:
[1] |
J. Bao, A. Truman and C. Yuan,
Almost sure asymptotic stability of stochastic partial differential equations with jumps, SIAM J. Control Optim., 49 (2011), 771-787.
doi: 10.1137/100786812. |
[2] |
A. Bensoussan, G. D. Prato, M.C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, Springer Science & Business Media, 2007.
doi: 10.1007/978-0-8176-4581-6. |
[3] |
G. Da Prato and J. Zabczyk, Stochastic Differentical Equations in Infinite Dimensions, Cambridge University Press, Cambridge, UK, 1992.
doi: 10.1017/CBO9780511666223.![]() ![]() ![]() |
[4] |
S. Dashkovskiy and M. Kosmykov,
Input-to-state stability of interconnected hybrid systems, Automatica, 49 (2013), 1068-1074.
doi: 10.1016/j.automatica.2013.01.045. |
[5] |
Y. Guo, W. Zhao and X. Ding,
Input-to-state stability for stochastic multi-group models with multi-dispersal and time-varying delay, Appl. Math. Comput., 343 (2019), 114-127.
doi: 10.1016/j.amc.2018.07.058. |
[6] |
B. Gess and J. M. Tölle,
Stability of solutions to stochastic partial differential equations, J. Differ. Equations, 260 (2016), 4973-5025.
doi: 10.1016/j.jde.2015.11.039. |
[7] |
T. E. Govindan and N. U. Ahmed,
Robust stabilization with a general decay of mild solutions of stochastic evolution equations, Stat. Probab. Lett., 83 (2013), 115-122.
doi: 10.1016/j.spl.2012.08.019. |
[8] |
T. E. Govindan and N. U. Ahmed,
A note on exponential state feedback stabilizability by a Razumikhin type theorem of mild solutions of SDEs with delay, Stat. Probab. Lett., 82 (2012), 1303-1309.
doi: 10.1016/j.spl.2012.03.027. |
[9] |
H. Ito and Y. Nishimura, An iISS framework for stochastic robustness of interconnected nonlinear systems, IEEE Trans. Autom. Control, 61 (2016), 1508-1523.
doi: 10.1109/TAC.2015.2471777. |
[10] |
H. Ito,
A complete characterization of integral input-to-state stability and its small-gain theorem for stochastic systems, IEEE Trans. Autom. Control, 65 (2020), 3039-3052.
doi: 10.1109/TAC.2019.2946203. |
[11] |
Y. Kang, D. Zhai, G. Liu, Y. Zhao and P. Zhao,
Stability analysis of a class of hybrid stochastic retarded systems under asynchronous switching, IEEE Trans. Autom. Control, 59 (2014), 1511-1523.
doi: 10.1109/TAC.2014.2305931. |
[12] |
S.-J. Liu, J.-F. Zhang and Z.-P. Jiang,
A notion of stochastic input-to-state stability and its application to stability of cascaded stochastic nonlinear systems, Acta Math. Appl. Sin.-Engl. Ser., 24 (2008), 141-156.
doi: 10.1007/s10255-007-7005-x. |
[13] |
J. Luo,
Stability of stochastic partial differential equations with infinite delays, J. Comput. Appl. Math., 222 (2008), 364-371.
doi: 10.1016/j.cam.2007.11.002. |
[14] |
K. Liu, Stability of Infinite Dimensional Stochastic Differential Equations with Applications, Chapman Hall, CRC, London, 2006. |
[15] |
J. Luo and K. Liu,
Stability of infinite dimensional stochastic evolution equations with memory and Markovian jumps, Stoch. Process. Their Appl., 118 (2008), 864-895.
doi: 10.1016/j.spa.2007.06.009. |
[16] |
M. Y. Li and Z. Shuai,
Global-stability problem for coupled systems of differential equations on networks, J. Differ. Equ., 248 (2010), 1-20.
doi: 10.1016/j.jde.2009.09.003. |
[17] |
A. Mironchenko,
Criteria for input-to-state practical stability, IEEE Trans. Autom. Control, 64 (2019), 298-304.
doi: 10.1109/TAC.2018.2824983. |
[18] |
A. Mironchenko and F. Wirth,
Monotonicity methods for input-to-state stability of nonlinear parabolic PDEs with boundary disturbances, SIAM J. Control Optim., 57 (2019), 510-532.
doi: 10.1137/17M1161877. |
[19] |
A. Mironchenko and H. Ito,
Characterizations of integral input-to-state stability for bilinear systems in infinite dimensions, Math. Control Relat. Fields, 6 (2016), 447-466.
doi: 10.3934/mcrf.2016011. |
[20] |
A. Mironchenko and F. Wirth,
Characterizations of input-to-state stability for infinite-dimensional systems, IEEE Trans. Autom. Control, 63 (2018), 1692-1707.
doi: 10.1109/tac.2017.2756341. |
[21] |
A. Mironchenko and F. Wirth,
Lyapunov characterization of input-to-state stability for semilinear control systems over Banach spaces, Syst. Control Lett., 119 (2018), 64-70.
doi: 10.1016/j.sysconle.2018.07.007. |
[22] |
R. Nabiullin, Input-to-State Stability and Stabilizability of Infinite-Dimensional Linear Systems, Diss. Universität Wuppertal, Fakultät für Mathematik und Naturwissenschaften Mathematik und Informatik Dissertationen, 2018. |
[23] |
S. Peng and F. Deng,
New criteria on pth moment input-to-state stability of impulsive stochastic delayed differential systems, IEEE Trans. Autom. Control, 62 (2017), 3573-3579.
doi: 10.1109/TAC.2017.2660066. |
[24] |
W. Ren and J. Xiong,
Stability analysis of impulsive stochastic nonlinear systems, IEEE Trans. Autom. Control, 62 (2017), 4791-4797.
doi: 10.1109/TAC.2017.2688350. |
[25] |
E. D. Sontag,
Smooth stabilization implies coprime factorization, IEEE Trans. Autom. Control, 34 (1989), 435-443.
doi: 10.1109/9.28018. |
[26] |
E. D. Sontag,
Comments on integral variants of ISS, Syst. Control Lett., 34 (1998), 93-100.
doi: 10.1016/S0167-6911(98)00003-6. |
[27] |
A. R. Teel, A. Subbaraman and A. Sferlazza,
Stability analysis for stochastic hybrid systems: A survey, Automatica, 50 (2014), 2435-2456.
doi: 10.1016/j.automatica.2014.08.006. |
[28] |
T. Taniguchi, K. Liu and A. Truman,
Existence, uniqueness and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces, J. Differ. Equations, 181 (2002), 72-91.
doi: 10.1006/jdeq.2001.4073. |
[29] |
X. Wu, Y. Tang and W. Zhang,
Input-to-state stability of impulsive stochastic delayed systems under linear assumptions, Automatica, 66 (2016), 195-204.
doi: 10.1016/j.automatica.2016.01.002. |
[30] |
P. Zhao, W. Feng and Y. Kang,
Stochastic input-to-state stability of switched stochastic nonlinear systems, Automatica, 48 (2012), 2569-2576.
doi: 10.1016/j.automatica.2012.06.058. |
show all references
References:
[1] |
J. Bao, A. Truman and C. Yuan,
Almost sure asymptotic stability of stochastic partial differential equations with jumps, SIAM J. Control Optim., 49 (2011), 771-787.
doi: 10.1137/100786812. |
[2] |
A. Bensoussan, G. D. Prato, M.C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, Springer Science & Business Media, 2007.
doi: 10.1007/978-0-8176-4581-6. |
[3] |
G. Da Prato and J. Zabczyk, Stochastic Differentical Equations in Infinite Dimensions, Cambridge University Press, Cambridge, UK, 1992.
doi: 10.1017/CBO9780511666223.![]() ![]() ![]() |
[4] |
S. Dashkovskiy and M. Kosmykov,
Input-to-state stability of interconnected hybrid systems, Automatica, 49 (2013), 1068-1074.
doi: 10.1016/j.automatica.2013.01.045. |
[5] |
Y. Guo, W. Zhao and X. Ding,
Input-to-state stability for stochastic multi-group models with multi-dispersal and time-varying delay, Appl. Math. Comput., 343 (2019), 114-127.
doi: 10.1016/j.amc.2018.07.058. |
[6] |
B. Gess and J. M. Tölle,
Stability of solutions to stochastic partial differential equations, J. Differ. Equations, 260 (2016), 4973-5025.
doi: 10.1016/j.jde.2015.11.039. |
[7] |
T. E. Govindan and N. U. Ahmed,
Robust stabilization with a general decay of mild solutions of stochastic evolution equations, Stat. Probab. Lett., 83 (2013), 115-122.
doi: 10.1016/j.spl.2012.08.019. |
[8] |
T. E. Govindan and N. U. Ahmed,
A note on exponential state feedback stabilizability by a Razumikhin type theorem of mild solutions of SDEs with delay, Stat. Probab. Lett., 82 (2012), 1303-1309.
doi: 10.1016/j.spl.2012.03.027. |
[9] |
H. Ito and Y. Nishimura, An iISS framework for stochastic robustness of interconnected nonlinear systems, IEEE Trans. Autom. Control, 61 (2016), 1508-1523.
doi: 10.1109/TAC.2015.2471777. |
[10] |
H. Ito,
A complete characterization of integral input-to-state stability and its small-gain theorem for stochastic systems, IEEE Trans. Autom. Control, 65 (2020), 3039-3052.
doi: 10.1109/TAC.2019.2946203. |
[11] |
Y. Kang, D. Zhai, G. Liu, Y. Zhao and P. Zhao,
Stability analysis of a class of hybrid stochastic retarded systems under asynchronous switching, IEEE Trans. Autom. Control, 59 (2014), 1511-1523.
doi: 10.1109/TAC.2014.2305931. |
[12] |
S.-J. Liu, J.-F. Zhang and Z.-P. Jiang,
A notion of stochastic input-to-state stability and its application to stability of cascaded stochastic nonlinear systems, Acta Math. Appl. Sin.-Engl. Ser., 24 (2008), 141-156.
doi: 10.1007/s10255-007-7005-x. |
[13] |
J. Luo,
Stability of stochastic partial differential equations with infinite delays, J. Comput. Appl. Math., 222 (2008), 364-371.
doi: 10.1016/j.cam.2007.11.002. |
[14] |
K. Liu, Stability of Infinite Dimensional Stochastic Differential Equations with Applications, Chapman Hall, CRC, London, 2006. |
[15] |
J. Luo and K. Liu,
Stability of infinite dimensional stochastic evolution equations with memory and Markovian jumps, Stoch. Process. Their Appl., 118 (2008), 864-895.
doi: 10.1016/j.spa.2007.06.009. |
[16] |
M. Y. Li and Z. Shuai,
Global-stability problem for coupled systems of differential equations on networks, J. Differ. Equ., 248 (2010), 1-20.
doi: 10.1016/j.jde.2009.09.003. |
[17] |
A. Mironchenko,
Criteria for input-to-state practical stability, IEEE Trans. Autom. Control, 64 (2019), 298-304.
doi: 10.1109/TAC.2018.2824983. |
[18] |
A. Mironchenko and F. Wirth,
Monotonicity methods for input-to-state stability of nonlinear parabolic PDEs with boundary disturbances, SIAM J. Control Optim., 57 (2019), 510-532.
doi: 10.1137/17M1161877. |
[19] |
A. Mironchenko and H. Ito,
Characterizations of integral input-to-state stability for bilinear systems in infinite dimensions, Math. Control Relat. Fields, 6 (2016), 447-466.
doi: 10.3934/mcrf.2016011. |
[20] |
A. Mironchenko and F. Wirth,
Characterizations of input-to-state stability for infinite-dimensional systems, IEEE Trans. Autom. Control, 63 (2018), 1692-1707.
doi: 10.1109/tac.2017.2756341. |
[21] |
A. Mironchenko and F. Wirth,
Lyapunov characterization of input-to-state stability for semilinear control systems over Banach spaces, Syst. Control Lett., 119 (2018), 64-70.
doi: 10.1016/j.sysconle.2018.07.007. |
[22] |
R. Nabiullin, Input-to-State Stability and Stabilizability of Infinite-Dimensional Linear Systems, Diss. Universität Wuppertal, Fakultät für Mathematik und Naturwissenschaften Mathematik und Informatik Dissertationen, 2018. |
[23] |
S. Peng and F. Deng,
New criteria on pth moment input-to-state stability of impulsive stochastic delayed differential systems, IEEE Trans. Autom. Control, 62 (2017), 3573-3579.
doi: 10.1109/TAC.2017.2660066. |
[24] |
W. Ren and J. Xiong,
Stability analysis of impulsive stochastic nonlinear systems, IEEE Trans. Autom. Control, 62 (2017), 4791-4797.
doi: 10.1109/TAC.2017.2688350. |
[25] |
E. D. Sontag,
Smooth stabilization implies coprime factorization, IEEE Trans. Autom. Control, 34 (1989), 435-443.
doi: 10.1109/9.28018. |
[26] |
E. D. Sontag,
Comments on integral variants of ISS, Syst. Control Lett., 34 (1998), 93-100.
doi: 10.1016/S0167-6911(98)00003-6. |
[27] |
A. R. Teel, A. Subbaraman and A. Sferlazza,
Stability analysis for stochastic hybrid systems: A survey, Automatica, 50 (2014), 2435-2456.
doi: 10.1016/j.automatica.2014.08.006. |
[28] |
T. Taniguchi, K. Liu and A. Truman,
Existence, uniqueness and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces, J. Differ. Equations, 181 (2002), 72-91.
doi: 10.1006/jdeq.2001.4073. |
[29] |
X. Wu, Y. Tang and W. Zhang,
Input-to-state stability of impulsive stochastic delayed systems under linear assumptions, Automatica, 66 (2016), 195-204.
doi: 10.1016/j.automatica.2016.01.002. |
[30] |
P. Zhao, W. Feng and Y. Kang,
Stochastic input-to-state stability of switched stochastic nonlinear systems, Automatica, 48 (2012), 2569-2576.
doi: 10.1016/j.automatica.2012.06.058. |
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