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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 |
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.
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S. Bonaccori,
Stochastic variation of constants formular for infinite dimensional equation, Stochastic Analysis and Applications, 17 (1999), 509-528.
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R. Curtain and H. J. Zwart, An Introduction to Infinite Dimensional Linear Systems Theory, 2$^{nd}$ edition, Springer-Verlag, New York, 1995.
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Filting and LQG problems for discrete-time stochastic singular systems, IEEE Transactions on Automatic Control, 34 (1989), 1105-1108.
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Observer-based controller design for stochastic descriptor systems with Brownian motions, Automatica, 49 (2013), 2229-2235.
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B. Gashi and A. A. Pantelous,
Linear backward stochastic differential equations of descriptor type: Regular systems, Stochastic Analysis and Application, 31 (2013), 142-166.
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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. |
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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.
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Z. Q. Ge, G. 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. Ge, G. 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. |
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S. G. Hu, C. M. Huang and F. K. Wu, Stochastic Differential Equation, Science Press, Beijing, 2008.
![]() ![]() |
[14] |
K. F. Kong, Y. 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.
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[16] |
K. B. Liaskos, A. 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. Liaskos, A. A. Pantelous and I. G. Stratis,
Stochastic degenerate Sobolev equation: well posedness and exact controllability, Math. Meth. App. Sci., 41 (2018), 1025-1032.
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X. Mao, Stochastic Differential Equation and Their Applications, Horwood Publishing, England, 1998. |
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I. V. Melnikova, A. I. Filikov and U. A. Anufrieva,
Abstract stochastic equations. I. classical and distributional solutions, J. Math. Sciences, Functional Analysis, 111 (2002), 3430-3475.
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I. V. Melnikova and A. I. Filikov, Abstract Cauchy Problem, Chapnan and Hall/CRC, London, 2001.
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B. Oksendal, Stochastic Differential Equation: An Introduction with Application, Springer-Verlag, New York, 1998.
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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. Zhang, Q. Ma, B. Y. Zhang, S. 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. Zhang, Y. 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. Zhao, Y. C. Ma, A. H. Chen, L. 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. Ge, G. 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. Ge, G. 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. Hu, C. M. Huang and F. K. Wu, Stochastic Differential Equation, Science Press, Beijing, 2008.
![]() ![]() |
[14] |
K. F. Kong, Y. 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. Liaskos, A. 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. Liaskos, A. 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. Melnikova, A. 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. Zhang, Q. Ma, B. Y. Zhang, S. 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. Zhang, Y. 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. Zhao, Y. C. Ma, A. H. Chen, L. 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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