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June  2022, 12(2): 339-351. doi: 10.3934/naco.2021009

Controllability and observability of stochastic implicit systems and stochastic GE-evolution operator

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China

* Corresponding author: gezqjd@mail.xjtu.edu.cn

Received  October 2020 Revised  February 2021 Published  June 2022 Early access  March 2021

Fund Project: The author is supported by National Natural Science Foundation of China grant Nos. 11926402 and 61973338

This paper discusses exact (approximate) controllability and exact (approximate) observability of stochastic implicit systems in Banach spaces. Firstly, we introduce the stochastic GE-evolution operator in Banach space and discuss existence and uniqueness of the mild solution to stochastic implicit systems by stochastic GE-evolution operator in Banach space. Secondly, we discuss conditions for exact (approximate) controllability and exact (approximate) observability of the systems considered in terms of stochastic GE-evolution operator and the dual principle. Finally, an illustrative example is given.

Citation: Zhaoqiang Ge. Controllability and observability of stochastic implicit systems and stochastic GE-evolution operator. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 339-351. doi: 10.3934/naco.2021009
References:
[1]

S. Bonaccori, Stochastic variation of constants formular for infinite dimensional equation, Stochastic Analysis and Applications, 17 (1999), 509-528.  doi: 10.1080/07362999908809616.

[2]

R. Curtain and H. J. Zwart, An Introduction to Infinite Dimensional Linear Systems Theory, 2$^{nd}$ edition, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6.

[3]

L. Dai, Filting and LQG problems for discrete-time stochastic singular systems, IEEE Transactions on Automatic Control, 34 (1989), 1105-1108.  doi: 10.1109/9.35288.

[4]

Z. W. Gao and X. Y. Shi, Observer-based controller design for stochastic descriptor systems with Brownian motions, Automatica, 49 (2013), 2229-2235.  doi: 10.1016/j.automatica.2013.04.001.

[5]

B. Gashi and A. A. Pantelous, Linear backward stochastic differential equations of descriptor type: Regular systems, Stochastic Analysis and Application, 31 (2013), 142-166.  doi: 10.1080/07362994.2013.741400.

[6]

B. Gashi and A. A. Pantelous, Linear stochastic systems of descriptor type: theory and applications, safety, reliability, risk and life-cycle performance of structure and infrastructures, in: Proceedings of the 11th international conference on structure safety and reliability, ICOSSAR 2013, (2013), 1047–1054.

[7]

B. Gashi and A. A. Pantelous, Linear backward stochastic differential systems of descriptor type with structure and applications to engineering, Probabilitic Engineering Mechanics, 40 (2015), 1-11.  doi: 10.1080/07362994.2013.741400.

[8]

Z. Q. GeG. T. Zhu and D. X. Feng, Exact controllability for singular distributed parameter systems in Hilbert spaces, Sci. China Inf. Sci., 52 (2009), 2045-2052.  doi: 10.1007/s11432-009-0204-8.

[9]

Z. Q. GeG. T. Zhu and D. X. Feng, Generalized operator semigroup and well-posedness of singular distributed parameter systems, Sci. Sin. Math., 40 (2010), 477-495. 

[10]

Z. Q. Ge, and D. X. Feng, Well-posed problem of nonlinear singular distributed parameter systems and nonlinear GE-semigroups, Sci. China Inf. Sci., 56 (2013), 128201: 1–128201: 14. doi: 10.1007/s11432-013-4852-3.

[11]

Z. Q. Ge and X. C. Ge, An exact controllability of stochastic singular systems, Sci. China Inf. Sci., 64 (2021), 179202: 1–179202: 3. doi: 10.1007/s11432-019-9902-y.

[12]

Z. Q. Ge, Impulse controllability and impulse observability of stochastic singular systems, J. Syst. Sci. Complex, 2020. doi: 10.1007/s11424-020-9250-5.

[13] S. G. HuC. M. Huang and F. K. Wu, Stochastic Differential Equation, Science Press, Beijing, 2008. 
[14]

K. F. KongY. C. Ma and D. Y. Liu, Observer-based quantized sliding mode dissipative control for singular semi-Markovian jump systems, Applied Mathematics and Computation, 362 (2019), 1-18.  doi: 10.1016/j.amc.2019.06.053.

[15]

K. L. Kuttler and J. Li, Generalized stochastic evolution equations, J. Differential Equations, 257 (2014), 816-842.  doi: 10.1016/j.jde.2014.04.017.

[16]

K. B. LiaskosA. A. Pantelous and I. G. Stratis, Linear stochastic degenerate Sobolev equation and application, International Journal of Control, 88 (2015), 2538-2553.  doi: 10.1080/00207179.2015.1048482.

[17]

K. B. LiaskosA. A. Pantelous and I. G. Stratis, Stochastic degenerate Sobolev equation: well posedness and exact controllability, Math. Meth. App. Sci., 41 (2018), 1025-1032.  doi: 10.1002/mma.4077.

[18]

X. Mao, Stochastic Differential Equation and Their Applications, Horwood Publishing, England, 1998.

[19]

I. V. MelnikovaA. I. Filikov and U. A. Anufrieva, Abstract stochastic equations. I. classical and distributional solutions, J. Math. Sciences, Functional Analysis, 111 (2002), 3430-3475.  doi: 10.1023/A:1016006127598.

[20]

I. V. Melnikova and A. I. Filikov, Abstract Cauchy Problem, Chapnan and Hall/CRC, London, 2001. doi: 10.1201/9781420035490.

[21]

B. Oksendal, Stochastic Differential Equation: An Introduction with Application, Springer-Verlag, New York, 1998. doi: 10.1007/978-3-662-03620-4.

[22] G. D. Prato and J. Zabczyk, Stochastic Equation in Infinite Dimensions, 2$^{nd}$ edition, Cambridge University Press, London, 2014.  doi: 10.1017/CBO9781107295513.
[23]

L. A. Vlasenko and A. G. Rutkas, Stochastic impulse control of parabolic systems of Sobolev type, Differential Equations, 47 (2011), 1498-1507.  doi: 10.1134/S0012266111100132.

[24]

S. Y. Xing and Q. L. Zhang, Stability and exact observability of discrete stochastic singular systems based on generalized Lyapunov equations, IET Control Theory and Applications, 10 (2016), 971-980.  doi: 10.1049/iet-cta.2015.0896.

[25]

P. Yu and Y. C. Ma, Observer-based asynchronous control for Markov jump systems, Applied Mathematics and Computation, 377 (2020), 1-14.  doi: 10.1016/j.amc.2020.125184.

[26]

Q. L. Zhang, L. Li and X. G. Yan, etc, Sliding mode control for singular stochastic Markovian jump systems with uncertainties, Automatica, 79 (2017), 27-34. doi: 10.1016/j.automatica.2017.01.002.

[27]

G. M. ZhangQ. MaB. Y. ZhangS. Y. Xu and J. W. Xia, Admissibility and stabilization of stochastic singular Markovian jump systems with time delays, Systems and Control Letters, 114 (2018), 1-10.  doi: 10.1016/j.sysconle.2018.02.004.

[28]

W. H. ZhangY. Zhao and L. Sheng, Some remarks on stability of stochastic singular systems with state-dependent noise, Automatica, 51 (2015), 273-277.  doi: 10.1016/j.automatica.2014.10.044.

[29]

W. Y. ZhaoY. C. MaA. H. ChenL. Fu and Y. T. Zhang, Robust sliding mode control for Markovian jump singular systems with randomly changing structure, Applied Mathematics and Computation, 349 (2019), 81-96.  doi: 10.1016/j.amc.2018.12.014.

[30]

Y. Zhao and W. H. Zhang, New results on stability of singular stochastic Markov jump systems with state-dependent noise, Int. J. Robust Nonlinear Control, 26 (2016), 2169-2186.  doi: 10.1002/rnc.3401.

show all references

References:
[1]

S. Bonaccori, Stochastic variation of constants formular for infinite dimensional equation, Stochastic Analysis and Applications, 17 (1999), 509-528.  doi: 10.1080/07362999908809616.

[2]

R. Curtain and H. J. Zwart, An Introduction to Infinite Dimensional Linear Systems Theory, 2$^{nd}$ edition, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6.

[3]

L. Dai, Filting and LQG problems for discrete-time stochastic singular systems, IEEE Transactions on Automatic Control, 34 (1989), 1105-1108.  doi: 10.1109/9.35288.

[4]

Z. W. Gao and X. Y. Shi, Observer-based controller design for stochastic descriptor systems with Brownian motions, Automatica, 49 (2013), 2229-2235.  doi: 10.1016/j.automatica.2013.04.001.

[5]

B. Gashi and A. A. Pantelous, Linear backward stochastic differential equations of descriptor type: Regular systems, Stochastic Analysis and Application, 31 (2013), 142-166.  doi: 10.1080/07362994.2013.741400.

[6]

B. Gashi and A. A. Pantelous, Linear stochastic systems of descriptor type: theory and applications, safety, reliability, risk and life-cycle performance of structure and infrastructures, in: Proceedings of the 11th international conference on structure safety and reliability, ICOSSAR 2013, (2013), 1047–1054.

[7]

B. Gashi and A. A. Pantelous, Linear backward stochastic differential systems of descriptor type with structure and applications to engineering, Probabilitic Engineering Mechanics, 40 (2015), 1-11.  doi: 10.1080/07362994.2013.741400.

[8]

Z. Q. GeG. T. Zhu and D. X. Feng, Exact controllability for singular distributed parameter systems in Hilbert spaces, Sci. China Inf. Sci., 52 (2009), 2045-2052.  doi: 10.1007/s11432-009-0204-8.

[9]

Z. Q. GeG. T. Zhu and D. X. Feng, Generalized operator semigroup and well-posedness of singular distributed parameter systems, Sci. Sin. Math., 40 (2010), 477-495. 

[10]

Z. Q. Ge, and D. X. Feng, Well-posed problem of nonlinear singular distributed parameter systems and nonlinear GE-semigroups, Sci. China Inf. Sci., 56 (2013), 128201: 1–128201: 14. doi: 10.1007/s11432-013-4852-3.

[11]

Z. Q. Ge and X. C. Ge, An exact controllability of stochastic singular systems, Sci. China Inf. Sci., 64 (2021), 179202: 1–179202: 3. doi: 10.1007/s11432-019-9902-y.

[12]

Z. Q. Ge, Impulse controllability and impulse observability of stochastic singular systems, J. Syst. Sci. Complex, 2020. doi: 10.1007/s11424-020-9250-5.

[13] S. G. HuC. M. Huang and F. K. Wu, Stochastic Differential Equation, Science Press, Beijing, 2008. 
[14]

K. F. KongY. C. Ma and D. Y. Liu, Observer-based quantized sliding mode dissipative control for singular semi-Markovian jump systems, Applied Mathematics and Computation, 362 (2019), 1-18.  doi: 10.1016/j.amc.2019.06.053.

[15]

K. L. Kuttler and J. Li, Generalized stochastic evolution equations, J. Differential Equations, 257 (2014), 816-842.  doi: 10.1016/j.jde.2014.04.017.

[16]

K. B. LiaskosA. A. Pantelous and I. G. Stratis, Linear stochastic degenerate Sobolev equation and application, International Journal of Control, 88 (2015), 2538-2553.  doi: 10.1080/00207179.2015.1048482.

[17]

K. B. LiaskosA. A. Pantelous and I. G. Stratis, Stochastic degenerate Sobolev equation: well posedness and exact controllability, Math. Meth. App. Sci., 41 (2018), 1025-1032.  doi: 10.1002/mma.4077.

[18]

X. Mao, Stochastic Differential Equation and Their Applications, Horwood Publishing, England, 1998.

[19]

I. V. MelnikovaA. I. Filikov and U. A. Anufrieva, Abstract stochastic equations. I. classical and distributional solutions, J. Math. Sciences, Functional Analysis, 111 (2002), 3430-3475.  doi: 10.1023/A:1016006127598.

[20]

I. V. Melnikova and A. I. Filikov, Abstract Cauchy Problem, Chapnan and Hall/CRC, London, 2001. doi: 10.1201/9781420035490.

[21]

B. Oksendal, Stochastic Differential Equation: An Introduction with Application, Springer-Verlag, New York, 1998. doi: 10.1007/978-3-662-03620-4.

[22] G. D. Prato and J. Zabczyk, Stochastic Equation in Infinite Dimensions, 2$^{nd}$ edition, Cambridge University Press, London, 2014.  doi: 10.1017/CBO9781107295513.
[23]

L. A. Vlasenko and A. G. Rutkas, Stochastic impulse control of parabolic systems of Sobolev type, Differential Equations, 47 (2011), 1498-1507.  doi: 10.1134/S0012266111100132.

[24]

S. Y. Xing and Q. L. Zhang, Stability and exact observability of discrete stochastic singular systems based on generalized Lyapunov equations, IET Control Theory and Applications, 10 (2016), 971-980.  doi: 10.1049/iet-cta.2015.0896.

[25]

P. Yu and Y. C. Ma, Observer-based asynchronous control for Markov jump systems, Applied Mathematics and Computation, 377 (2020), 1-14.  doi: 10.1016/j.amc.2020.125184.

[26]

Q. L. Zhang, L. Li and X. G. Yan, etc, Sliding mode control for singular stochastic Markovian jump systems with uncertainties, Automatica, 79 (2017), 27-34. doi: 10.1016/j.automatica.2017.01.002.

[27]

G. M. ZhangQ. MaB. Y. ZhangS. Y. Xu and J. W. Xia, Admissibility and stabilization of stochastic singular Markovian jump systems with time delays, Systems and Control Letters, 114 (2018), 1-10.  doi: 10.1016/j.sysconle.2018.02.004.

[28]

W. H. ZhangY. Zhao and L. Sheng, Some remarks on stability of stochastic singular systems with state-dependent noise, Automatica, 51 (2015), 273-277.  doi: 10.1016/j.automatica.2014.10.044.

[29]

W. Y. ZhaoY. C. MaA. H. ChenL. Fu and Y. T. Zhang, Robust sliding mode control for Markovian jump singular systems with randomly changing structure, Applied Mathematics and Computation, 349 (2019), 81-96.  doi: 10.1016/j.amc.2018.12.014.

[30]

Y. Zhao and W. H. Zhang, New results on stability of singular stochastic Markov jump systems with state-dependent noise, Int. J. Robust Nonlinear Control, 26 (2016), 2169-2186.  doi: 10.1002/rnc.3401.

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