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February  2022, 27(2): 821-836. doi: 10.3934/dcdsb.2021066

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

Pengfei Wang , Mengyi Zhang and  Huan Su

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

Received  July 2020 Revised  December 2020 Published  February 2022 Early access  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.

Keywords: Infinite-dimensional stochastic systems, mild solutions, input-to-state stability, Lyapunov method, Itô's formula.
Mathematics Subject Classification: Primary: 37C75; Secondary: 60H15, 93D30.
Citation: Pengfei Wang, Mengyi Zhang, Huan Su. Input-to-state stability of infinite-dimensional stochastic nonlinear systems. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 821-836. doi: 10.3934/dcdsb.2021066
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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