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June  2020, 10(2): 443-454. doi: 10.3934/mcrf.2020005

Switching controls for linear stochastic differential systems

Yong He ,

School of Mathematics, Physics and Data Science, Chongqing University of Science and Technology, Chongqing, 401331, China

* Corresponding author: Yong He

Received  May 2019 Revised  September 2019 Published  June 2020 Early access  November 2019

Fund Project: The author is supported by the science and technology research project of Chongqing Education Commission under grant KJQN201801529

We analyze the exact controllability problem of switching controls for stochastic control systems endowed with different actuators. The goal is to control the dynamics of the system by switching from an actuator to the other such that, in each instant of time, there are as few active actuators as possible. We prove that, under suitable rank conditions, switching control strategies exist and can be built in a systematic way. The proof is based on building a new functional by the adjoint system whose minimizers are the switching controls.

Keywords: Stochastic control system, switching control, exact controllability.
Mathematics Subject Classification: Primary: 93B05.
Citation: Yong He. Switching controls for linear stochastic differential systems. Mathematical Control & Related Fields, 2020, 10 (2) : 443-454. doi: 10.3934/mcrf.2020005
References:
[1]

G. Battistelli, J. Hespanha, E. Mosca and P. Tesi, Model-free adaptive switching control of time-varying plants, IEEE Trans. Automat. Control, 58 (2013), 1208-1220.  doi: 10.1109/TAC.2013.2243974.  Google Scholar

[2]

R. Buckdahn, M. Quincampoix and G. Tessitore, A characterization of approximately controllable linear stochastic differential equations, in Stochastic Partial Differential Equations and Applications–VII, Lect. Notes Pure Appl. Math., 245, Chapman & Hall/CRC, Boca Raton, FL, 2006, 53–60.  Google Scholar

[3]

C. Clason, A. Rund, K. Kunisch and R. C. Barnard, A convex penalty for switching control of partial differential equations, Systems Control Lett., 89 (2016), 66-73.  doi: 10.1016/j.sysconle.2015.12.013.  Google Scholar

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F. Dou and Q. Lu, Partial approximate controllability for linear stochastic control systems, SIAM J. Control Optim., 57 (2019), 1209-1229.  doi: 10.1137/18M1164640.  Google Scholar

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X. Fu and X. Liu, Controllability and observability of some stochastic complex Ginzburg-Landau equations, SIAM J. Control Optim., 55 (2017), 1102-1127.  doi: 10.1137/15M1039961.  Google Scholar

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A. Garulli, A. Giannitrapani and M. Leomanni, Minimum switching control for systems of coupled double integrators, Automatica J. IFAC, 60 (2015), 115-121.  doi: 10.1016/j.automatica.2015.07.004.  Google Scholar

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B. Gashi, Stochastic minimum-energy control, Systems Control Lett., 85 (2015), 70-76.  doi: 10.1016/j.sysconle.2015.08.012.  Google Scholar

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D. Goreac, A Kalman-type condition for stochastic approximate controllability, C. R. Math. Acad. Sci. Paris, 346 (2008), 183-188.  doi: 10.1016/j.crma.2007.12.008.  Google Scholar

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X. Liu, Controllability of some coupled stochastic parabolic systems with fractional order spatial differential operators by one control in the drift, SIAM J. Control Optim., 52 (2014), 836-860.  doi: 10.1137/130926791.  Google Scholar

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F. Liu and S. Peng, On controllability for stochastic control systems when the coefficient is time-variant, J. Syst. Sci. Complex., 23 (2010), 270-278.  doi: 10.1007/s11424-010-8158-x.  Google Scholar

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Q. Lü, Some results on the controllability of forward stochastic heat equations with control on the drift, J. Funct. Anal., 260 (2011), 832-851.  doi: 10.1016/j.jfa.2010.10.018.  Google Scholar

[12]

Q. Lü, Exact controllability for stochastic Schrödinger equations, J. Differential Equations, 255 (2013), 2484-2504.  doi: 10.1016/j.jde.2013.06.021.  Google Scholar

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Q. Lü, Exact controllability for stochastic transport equations, SIAM J. Control Optim., 52 (2014), 397-419.  doi: 10.1137/130910373.  Google Scholar

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Q. Lü, J. Yong and X. Zhang, Representation of Itô integrals by Lebesgue/Bochner integrals, J. Eur. Math. Soc., 14 (2012), 1795-1823.  doi: 10.4171/JEMS/347.  Google Scholar

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Q. Lü and E. Zuazua, Robust null controllability for heat equations with unknown switching control mode, Discrete Contin. Dyn. Syst., 34 (2014), 4183-4210.  doi: 10.3934/dcds.2014.34.4183.  Google Scholar

[16]

M. Ouzahra, Global stabilization of semilinear systems using switching controls, Automatica J. IFAC, 48 (2012), 837-843.  doi: 10.1016/j.automatica.2012.02.018.  Google Scholar

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S. Peng, Backward stochastic differential equation and exact controllability of stochastic control systems, Progr. Natur. Sci., 4 (1994), 274-284.   Google Scholar

[18]

R. Shorten, F. Wirth, O. Mason, K. Wulff and C. King, Stability criteria for switched and hybrid systems, SIAM Rev., 49 (2007), 545-592.  doi: 10.1137/05063516X.  Google Scholar

[19]

M. M. Tousi, I. Karuei, S. Hashtrudi-Zad and A. G. Aghdam, Supervisory control of switching control systems, Systems Control Lett., 57 (2008), 132-141.  doi: 10.1016/j.sysconle.2007.08.002.  Google Scholar

[20]

Y. Wang, BSDEs with general filtration driven by Lévy processes, and an application in stochastic controllability, Systems Control Lett., 62 (2013), 242-247.  doi: 10.1016/j.sysconle.2012.11.021.  Google Scholar

[21]

Y. Wang, D. Yang, J. Yong and Z. Yu, Exact controllability of linear stochastic differential equations and related problems, Math. Control Relat. Fields, 7 (2017), 305-345.  doi: 10.3934/mcrf.2017011.  Google Scholar

[22]

J. Yong and X. Zhou, Stochastic Controls. Hamiltonian Systems and HJB Equations, Applications of Mathematics, 43, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.  Google Scholar

[23]

W. Zhang and B.-S. Chen, On stabilizability and exact observability of stochastic systems with their applications, Automatica J. IFAC, 40 (2004), 87-94.  doi: 10.1016/j.automatica.2003.07.002.  Google Scholar

[24]

E. Zuazua, Switching control, J. Eur. Math. Soc. (JEMS), 13 (2011), 85-117.  doi: 10.4171/JEMS/245.  Google Scholar

show all references

References:
[1]

G. Battistelli, J. Hespanha, E. Mosca and P. Tesi, Model-free adaptive switching control of time-varying plants, IEEE Trans. Automat. Control, 58 (2013), 1208-1220.  doi: 10.1109/TAC.2013.2243974.  Google Scholar

[2]

R. Buckdahn, M. Quincampoix and G. Tessitore, A characterization of approximately controllable linear stochastic differential equations, in Stochastic Partial Differential Equations and Applications–VII, Lect. Notes Pure Appl. Math., 245, Chapman & Hall/CRC, Boca Raton, FL, 2006, 53–60.  Google Scholar

[3]

C. Clason, A. Rund, K. Kunisch and R. C. Barnard, A convex penalty for switching control of partial differential equations, Systems Control Lett., 89 (2016), 66-73.  doi: 10.1016/j.sysconle.2015.12.013.  Google Scholar

[4]

F. Dou and Q. Lu, Partial approximate controllability for linear stochastic control systems, SIAM J. Control Optim., 57 (2019), 1209-1229.  doi: 10.1137/18M1164640.  Google Scholar

[5]

X. Fu and X. Liu, Controllability and observability of some stochastic complex Ginzburg-Landau equations, SIAM J. Control Optim., 55 (2017), 1102-1127.  doi: 10.1137/15M1039961.  Google Scholar

[6]

A. Garulli, A. Giannitrapani and M. Leomanni, Minimum switching control for systems of coupled double integrators, Automatica J. IFAC, 60 (2015), 115-121.  doi: 10.1016/j.automatica.2015.07.004.  Google Scholar

[7]

B. Gashi, Stochastic minimum-energy control, Systems Control Lett., 85 (2015), 70-76.  doi: 10.1016/j.sysconle.2015.08.012.  Google Scholar

[8]

D. Goreac, A Kalman-type condition for stochastic approximate controllability, C. R. Math. Acad. Sci. Paris, 346 (2008), 183-188.  doi: 10.1016/j.crma.2007.12.008.  Google Scholar

[9]

X. Liu, Controllability of some coupled stochastic parabolic systems with fractional order spatial differential operators by one control in the drift, SIAM J. Control Optim., 52 (2014), 836-860.  doi: 10.1137/130926791.  Google Scholar

[10]

F. Liu and S. Peng, On controllability for stochastic control systems when the coefficient is time-variant, J. Syst. Sci. Complex., 23 (2010), 270-278.  doi: 10.1007/s11424-010-8158-x.  Google Scholar

[11]

Q. Lü, Some results on the controllability of forward stochastic heat equations with control on the drift, J. Funct. Anal., 260 (2011), 832-851.  doi: 10.1016/j.jfa.2010.10.018.  Google Scholar

[12]

Q. Lü, Exact controllability for stochastic Schrödinger equations, J. Differential Equations, 255 (2013), 2484-2504.  doi: 10.1016/j.jde.2013.06.021.  Google Scholar

[13]

Q. Lü, Exact controllability for stochastic transport equations, SIAM J. Control Optim., 52 (2014), 397-419.  doi: 10.1137/130910373.  Google Scholar

[14]

Q. Lü, J. Yong and X. Zhang, Representation of Itô integrals by Lebesgue/Bochner integrals, J. Eur. Math. Soc., 14 (2012), 1795-1823.  doi: 10.4171/JEMS/347.  Google Scholar

[15]

Q. Lü and E. Zuazua, Robust null controllability for heat equations with unknown switching control mode, Discrete Contin. Dyn. Syst., 34 (2014), 4183-4210.  doi: 10.3934/dcds.2014.34.4183.  Google Scholar

[16]

M. Ouzahra, Global stabilization of semilinear systems using switching controls, Automatica J. IFAC, 48 (2012), 837-843.  doi: 10.1016/j.automatica.2012.02.018.  Google Scholar

[17]

S. Peng, Backward stochastic differential equation and exact controllability of stochastic control systems, Progr. Natur. Sci., 4 (1994), 274-284.   Google Scholar

[18]

R. Shorten, F. Wirth, O. Mason, K. Wulff and C. King, Stability criteria for switched and hybrid systems, SIAM Rev., 49 (2007), 545-592.  doi: 10.1137/05063516X.  Google Scholar

[19]

M. M. Tousi, I. Karuei, S. Hashtrudi-Zad and A. G. Aghdam, Supervisory control of switching control systems, Systems Control Lett., 57 (2008), 132-141.  doi: 10.1016/j.sysconle.2007.08.002.  Google Scholar

[20]

Y. Wang, BSDEs with general filtration driven by Lévy processes, and an application in stochastic controllability, Systems Control Lett., 62 (2013), 242-247.  doi: 10.1016/j.sysconle.2012.11.021.  Google Scholar

[21]

Y. Wang, D. Yang, J. Yong and Z. Yu, Exact controllability of linear stochastic differential equations and related problems, Math. Control Relat. Fields, 7 (2017), 305-345.  doi: 10.3934/mcrf.2017011.  Google Scholar

[22]

J. Yong and X. Zhou, Stochastic Controls. Hamiltonian Systems and HJB Equations, Applications of Mathematics, 43, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.  Google Scholar

[23]

W. Zhang and B.-S. Chen, On stabilizability and exact observability of stochastic systems with their applications, Automatica J. IFAC, 40 (2004), 87-94.  doi: 10.1016/j.automatica.2003.07.002.  Google Scholar

[24]

E. Zuazua, Switching control, J. Eur. Math. Soc. (JEMS), 13 (2011), 85-117.  doi: 10.4171/JEMS/245.  Google Scholar

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