doi: 10.3934/dcdsb.2021066

Input-to-state stability of infinite-dimensional stochastic nonlinear systems

Department of Mathematics, Harbin Institute of Technology at Weihai, Weihai 264209, PR China

Received  July 2020 Revised  December 2020 Published  March 2021

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.

Citation: Pengfei Wang, Mengyi Zhang, Huan Su. Input-to-state stability of infinite-dimensional stochastic nonlinear systems. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021066
References:
[1]

J. BaoA. 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.  Google Scholar

[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.  Google Scholar

[3] G. Da Prato and J. Zabczyk, Stochastic Differentical Equations in Infinite Dimensions, Cambridge University Press, Cambridge, UK, 1992.  doi: 10.1017/CBO9780511666223.  Google Scholar
[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.  Google Scholar

[5]

Y. GuoW. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

Y. KangD. ZhaiG. LiuY. 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.  Google Scholar

[12]

S.-J. LiuJ.-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.  Google Scholar

[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.  Google Scholar

[14]

K. Liu, Stability of Infinite Dimensional Stochastic Differential Equations with Applications, Chapman Hall, CRC, London, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

A. Mironchenko, Criteria for input-to-state practical stability, IEEE Trans. Autom. Control, 64 (2019), 298-304.  doi: 10.1109/TAC.2018.2824983.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Autom. Control, 34 (1989), 435-443.  doi: 10.1109/9.28018.  Google Scholar

[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.  Google Scholar

[27]

A. R. TeelA. 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.  Google Scholar

[28]

T. TaniguchiK. 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.  Google Scholar

[29]

X. WuY. 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.  Google Scholar

[30]

P. ZhaoW. 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.  Google Scholar

show all references

References:
[1]

J. BaoA. 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.  Google Scholar

[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.  Google Scholar

[3] G. Da Prato and J. Zabczyk, Stochastic Differentical Equations in Infinite Dimensions, Cambridge University Press, Cambridge, UK, 1992.  doi: 10.1017/CBO9780511666223.  Google Scholar
[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.  Google Scholar

[5]

Y. GuoW. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

Y. KangD. ZhaiG. LiuY. 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.  Google Scholar

[12]

S.-J. LiuJ.-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.  Google Scholar

[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.  Google Scholar

[14]

K. Liu, Stability of Infinite Dimensional Stochastic Differential Equations with Applications, Chapman Hall, CRC, London, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

A. Mironchenko, Criteria for input-to-state practical stability, IEEE Trans. Autom. Control, 64 (2019), 298-304.  doi: 10.1109/TAC.2018.2824983.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Autom. Control, 34 (1989), 435-443.  doi: 10.1109/9.28018.  Google Scholar

[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.  Google Scholar

[27]

A. R. TeelA. 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.  Google Scholar

[28]

T. TaniguchiK. 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.  Google Scholar

[29]

X. WuY. 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.  Google Scholar

[30]

P. ZhaoW. 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.  Google Scholar

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