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

Previous Article
Second order discrete time-varying and time-invariant linear continuous systems and Kalman type conditions
NACO Home
This Issue
Next Article
Applying particle swarm optimization based on Padé approximant to solve ordinary differential equation

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

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

Zhaoqiang Ge ^,

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.

Keywords: Stochastic implicit systems, controllability, observability, stochastic GE-evolution operator, Banach space.

Mathematics Subject Classification: Primary: 93B05, 93B07.

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. 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.

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.

[1]	Lingyang Liu, Xu Liu. Controllability and observability of some coupled stochastic parabolic systems. Mathematical Control and Related Fields, 2018, 8 (3&4) : 829-854. doi: 10.3934/mcrf.2018037
[2]	Daliang Zhao, Yansheng Liu. Controllability of nonlinear fractional evolution systems in Banach spaces: A survey. Electronic Research Archive, 2021, 29 (5) : 3551-3580. doi: 10.3934/era.2021083
[3]	Zdzisław Brzeźniak, Paul André Razafimandimby. Irreducibility and strong Feller property for stochastic evolution equations in Banach spaces. Discrete and Continuous Dynamical Systems - B, 2016, 21 (4) : 1051-1077. doi: 10.3934/dcdsb.2016.21.1051
[4]	Pengyu Chen, Xuping Zhang. Approximate controllability of nonlocal problem for non-autonomous stochastic evolution equations. Evolution Equations and Control Theory, 2021, 10 (3) : 471-489. doi: 10.3934/eect.2020076
[5]	Viorel Barbu, Gabriela Marinoschi. An identification problem for a linear evolution equation in a banach space. Discrete and Continuous Dynamical Systems - S, 2020, 13 (5) : 1429-1440. doi: 10.3934/dcdss.2020081
[6]	Tôn Việt Tạ. Non-autonomous stochastic evolution equations in Banach spaces of martingale type 2: Strict solutions and maximal regularity. Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4507-4542. doi: 10.3934/dcds.2017193
[7]	Tôn Việt Tạ. Strict solutions to stochastic semilinear evolution equations in M-type 2 Banach spaces. Communications on Pure and Applied Analysis, 2021, 20 (5) : 1867-1891. doi: 10.3934/cpaa.2021050
[8]	K. Ravikumar, Manil T. Mohan, A. Anguraj. Approximate controllability of a non-autonomous evolution equation in Banach spaces. Numerical Algebra, Control and Optimization, 2021, 11 (3) : 461-485. doi: 10.3934/naco.2020038
[9]	Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control and Related Fields, 2021, 11 (4) : 857-883. doi: 10.3934/mcrf.2020049
[10]	Qiang Du, Manlin Li. On the stochastic immersed boundary method with an implicit interface formulation. Discrete and Continuous Dynamical Systems - B, 2011, 15 (2) : 373-389. doi: 10.3934/dcdsb.2011.15.373
[11]	Alfredo Lorenzi, Ioan I. Vrabie. An identification problem for a linear evolution equation in a Banach space and applications. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 671-691. doi: 10.3934/dcdss.2011.4.671
[12]	Arzu Ahmadova, Nazim I. Mahmudov, Juan J. Nieto. Exponential stability and stabilization of fractional stochastic degenerate evolution equations in a Hilbert space: Subordination principle. Evolution Equations and Control Theory, 2022 doi: 10.3934/eect.2022008
[13]	Thuy N. T. Nguyen. Uniform controllability of semidiscrete approximations for parabolic systems in Banach spaces. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 613-640. doi: 10.3934/dcdsb.2015.20.613
[14]	Daoyi Xu, Weisong Zhou. Existence-uniqueness and exponential estimate of pathwise solutions of retarded stochastic evolution systems with time smooth diffusion coefficients. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 2161-2180. doi: 10.3934/dcds.2017093
[15]	Tatsien Li, Bopeng Rao, Zhiqiang Wang. Exact boundary controllability and observability for first order quasilinear hyperbolic systems with a kind of nonlocal boundary conditions. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 243-257. doi: 10.3934/dcds.2010.28.243
[16]	Andi Kivinukk, Anna Saksa. On Rogosinski-type approximation processes in Banach space using the framework of the cosine operator function. Mathematical Foundations of Computing, 2022, 5 (3) : 197-218. doi: 10.3934/mfc.2021030
[17]	Ankit Kumar, Kamal Jeet, Ramesh Kumar Vats. Controllability of Hilfer fractional integro-differential equations of Sobolev-type with a nonlocal condition in a Banach space. Evolution Equations and Control Theory, 2022, 11 (2) : 605-619. doi: 10.3934/eect.2021016
[18]	Tomás Caraballo, Leonid Shaikhet. Stability of delay evolution equations with stochastic perturbations. Communications on Pure and Applied Analysis, 2014, 13 (5) : 2095-2113. doi: 10.3934/cpaa.2014.13.2095
[19]	Anna Karczewska, Carlos Lizama. On stochastic fractional Volterra equations in Hilbert space. Conference Publications, 2007, 2007 (Special) : 541-550. doi: 10.3934/proc.2007.2007.541
[20]	Darko Volkov, Joan Calafell Sandiumenge. A stochastic approach to reconstruction of faults in elastic half space. Inverse Problems and Imaging, 2019, 13 (3) : 479-511. doi: 10.3934/ipi.2019024

Impact Factor:

Tools

Metrics

Other articles
by authors

on AIMS
Zhaoqiang Ge
on Google Scholar
Zhaoqiang Ge

Impact Factor:

Tools

Article outline

Impact Factor:

Tools

Metrics

Other articles
by authors

on AIMS
Zhaoqiang Ge
on Google Scholar
Zhaoqiang Ge

[Back to Top]