American Institute of Mathematical Sciences

Advanced
June  2017, 7(2): 305-345. doi: 10.3934/mcrf.2017011

Exact controllability of linear stochastic differential equations and related problems

Yanqing Wang 1, , Donghui Yang 2, , Jiongmin Yong 3, and  Zhiyong Yu 4,,

1. 

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
2. 

School of Mathematics and Statistics, School of Information Science and Engineering, Central South University, Changsha 410075, China
3. 

Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA
4. 

School of Mathematics, Shandong University, Jinan 250100, China

∗ Corresponding author: Zhiyong Yu.

Received  January 2017 Published  April 2017

Fund Project: the National Natural Science Foundation of China (11471192, 11371375, 11526167), the Fundamental Research Funds for the Central Universities (SWU113038, XDJK2014C076), the Nature Science Foundation of Shandong Province (JQ201401), the Natural Science Foundation of CQCSTC (2015jcyjA00017), China Postdoctoral Science Foundation and Central South University Postdoctoral Science Foundation, and NSF Grant DMS-1406776.

A notion of $L^p$-exact controllability is introduced for linear controlled (forward) stochastic differential equations with random coefficients. Several sufficient conditions are established for such kind of exact controllability. Further, it is proved that the $L^p$-exact controllability, the validity of an observability inequality for the adjoint equation, the solvability of an optimization problem, and the solvability of an $L^p$-type norm optimal control problem are all equivalent.

Keywords: Controlled stochastic differential equation, $L^p$-exact controllability, observability inequality, norm optimal control problem.
Mathematics Subject Classification: Primary: 93B05, 93E20; Secondary: 60H10.
Citation: Yanqing Wang, Donghui Yang, Jiongmin Yong, Zhiyong Yu. Exact controllability of linear stochastic differential equations and related problems. Mathematical Control and Related Fields, 2017, 7 (2) : 305-345. doi: 10.3934/mcrf.2017011
References:
[1]

R. Buckdahn, M. Quincampoix and G. Tessitore, A characterization of approximately controllable linear stochastic differential equations, , ().  doi: 10.1201/9781420028720.ch6.

[2]

S. Chen, X. Li, S. Peng and J. Yong, On stochastic linear controlled systems, Preprint, 1993.

[3]

M.M. Connors, Controllability of discrete, linear, random dynamical systems, SIAM J. Control, 5 (1967), 183-210.  doi: 10.1137/0305012.

[4]

E.D. Denman and A.N. Beavers,Jr., The matrix sign function and computations in systems, Appl. Math. Comput., 2 (1976), 63-94.  doi: 10.1016/0096-3003(76)90020-5.

[5]

M. Ehrhardt and W. Kliemann, Controllability of linear stochastic systems, , ().  doi: 10.1016/0167-6911(82)90012-3.

[6]

N. El Karoui, S. Peng and M.C. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), 1-71.  doi: 10.1111/1467-9965.00022.

[7]

H. O. Fattorini, Infinite-Dimensional Optimization and Control Theory Cambridge Univ. Press, Cambridge, 1999. doi: 10.1017/CBO9780511574795.

[8]

H.O. Fattorini, Time and norm optimal controls: A survey of recent results and open problems, Acta Math. Sci. Ser. B Engl. Ed., 31 (2011), 2203-2218.  doi: 10.1016/S0252-9602(11)60394-9.

[9]

G. B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd Edition John Wiley & Sons, New York, 1999.

[10]

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

[11]

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.

[12]

M. Gugat and G. Leugering, $L^∞$-norm minimal control of the wave equation: On the weakness of the bang-bang principle, ESAIM Control Optim. Calc. Var., 14 (2008), 254-283.  doi: 10.1051/cocv:2007044.

[13]

E. B. Lee and L. Markus, Foundations of Optimal Control Theory John Wiley & Sons, 1967.

[14]

J.L. Lions, Exact controllability, stabilizability and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68.  doi: 10.1137/1030001.

[15]

J. L. Lions, Controlabilité Exacte, Perturbations et Stabilisation de Systémes Distribués Masson, Paris, RMA 8,1988.

[16]

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.

[17]

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.

[18]

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

[19]

D.L. Russell, Controllability and stabilizability theory for linear partial differential equations, Recent Progress and open questions, SIAM Rev., 20 (1978), 639-739.  doi: 10.1137/1020095.

[20]

Y. Shi, T. Wang and J. Yong, Mean-field backward stochastoic Volterra integral equations, Discrete Continuous Dyn. Systems, Ser. B, 18 (2013), 1929-1967.  doi: 10.3934/dcdsb.2013.18.1929.

[21]

Y. Sunahara, S. Aihara and K. Kishino, On the stochastic observability and controllability for non-linear systems, Int. J Control, 22 (1975), 65-82.  doi: 10.1080/00207177508922061.

[22]

G. Wang and Y. Xu, Equivalence of three different kinds of optimal control problems for heat equations and its applications, SIAM J. Control Optim., 51 (2013), 848-880.  doi: 10.1137/110852449.

[23]

G. Wang and E. Zuazua, On the equivalence of minimal time and minimal norm controls for internally controlled heat equations, SIAM J. Control Optim., 50 (2012), 2938-2958.  doi: 10.1137/110857398.

[24]

Y. Wang and C. Zhang, The norm optimal control problem for stochastic linear control systems, ESAIM Control Optim. Calc. Var., 21 (2015), 399-413.  doi: 10.1051/cocv/2014030.

[25]

W. M. Wonham, Linear Multivariable Control: A Geometric Approach, 3rd Edition Springer Verlag, New York, 1985. doi: 10.1007/978-1-4612-1082-5.

[26]

J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations Springer, New York, 1999. doi: 10.1007/978-1-4612-1466-3.

[27]

K. Yosida, Functional Analysis Springer-Verlag, Berlin, New York, 1980.

[28]

J. Zabczyk, Controllability of stochastic linear systems, Systems Control Lett., 1 (1981), 25-31.  doi: 10.1016/S0167-6911(81)80008-4.

[29]

E. Zuazua, Controllability of Partial Differential Equations manuscript, 2006.

show all references

References:
[1]

R. Buckdahn, M. Quincampoix and G. Tessitore, A characterization of approximately controllable linear stochastic differential equations, , ().  doi: 10.1201/9781420028720.ch6.

[2]

S. Chen, X. Li, S. Peng and J. Yong, On stochastic linear controlled systems, Preprint, 1993.

[3]

M.M. Connors, Controllability of discrete, linear, random dynamical systems, SIAM J. Control, 5 (1967), 183-210.  doi: 10.1137/0305012.

[4]

E.D. Denman and A.N. Beavers,Jr., The matrix sign function and computations in systems, Appl. Math. Comput., 2 (1976), 63-94.  doi: 10.1016/0096-3003(76)90020-5.

[5]

M. Ehrhardt and W. Kliemann, Controllability of linear stochastic systems, , ().  doi: 10.1016/0167-6911(82)90012-3.

[6]

N. El Karoui, S. Peng and M.C. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), 1-71.  doi: 10.1111/1467-9965.00022.

[7]

H. O. Fattorini, Infinite-Dimensional Optimization and Control Theory Cambridge Univ. Press, Cambridge, 1999. doi: 10.1017/CBO9780511574795.

[8]

H.O. Fattorini, Time and norm optimal controls: A survey of recent results and open problems, Acta Math. Sci. Ser. B Engl. Ed., 31 (2011), 2203-2218.  doi: 10.1016/S0252-9602(11)60394-9.

[9]

G. B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd Edition John Wiley & Sons, New York, 1999.

[10]

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

[11]

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.

[12]

M. Gugat and G. Leugering, $L^∞$-norm minimal control of the wave equation: On the weakness of the bang-bang principle, ESAIM Control Optim. Calc. Var., 14 (2008), 254-283.  doi: 10.1051/cocv:2007044.

[13]

E. B. Lee and L. Markus, Foundations of Optimal Control Theory John Wiley & Sons, 1967.

[14]

J.L. Lions, Exact controllability, stabilizability and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68.  doi: 10.1137/1030001.

[15]

J. L. Lions, Controlabilité Exacte, Perturbations et Stabilisation de Systémes Distribués Masson, Paris, RMA 8,1988.

[16]

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.

[17]

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.

[18]

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

[19]

D.L. Russell, Controllability and stabilizability theory for linear partial differential equations, Recent Progress and open questions, SIAM Rev., 20 (1978), 639-739.  doi: 10.1137/1020095.

[20]

Y. Shi, T. Wang and J. Yong, Mean-field backward stochastoic Volterra integral equations, Discrete Continuous Dyn. Systems, Ser. B, 18 (2013), 1929-1967.  doi: 10.3934/dcdsb.2013.18.1929.

[21]

Y. Sunahara, S. Aihara and K. Kishino, On the stochastic observability and controllability for non-linear systems, Int. J Control, 22 (1975), 65-82.  doi: 10.1080/00207177508922061.

[22]

G. Wang and Y. Xu, Equivalence of three different kinds of optimal control problems for heat equations and its applications, SIAM J. Control Optim., 51 (2013), 848-880.  doi: 10.1137/110852449.

[23]

G. Wang and E. Zuazua, On the equivalence of minimal time and minimal norm controls for internally controlled heat equations, SIAM J. Control Optim., 50 (2012), 2938-2958.  doi: 10.1137/110857398.

[24]

Y. Wang and C. Zhang, The norm optimal control problem for stochastic linear control systems, ESAIM Control Optim. Calc. Var., 21 (2015), 399-413.  doi: 10.1051/cocv/2014030.

[25]

W. M. Wonham, Linear Multivariable Control: A Geometric Approach, 3rd Edition Springer Verlag, New York, 1985. doi: 10.1007/978-1-4612-1082-5.

[26]

J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations Springer, New York, 1999. doi: 10.1007/978-1-4612-1466-3.

[27]

K. Yosida, Functional Analysis Springer-Verlag, Berlin, New York, 1980.

[28]

J. Zabczyk, Controllability of stochastic linear systems, Systems Control Lett., 1 (1981), 25-31.  doi: 10.1016/S0167-6911(81)80008-4.

[29]

E. Zuazua, Controllability of Partial Differential Equations manuscript, 2006.

[1]

Luisa Malaguti, Stefania Perrotta, Valentina Taddei. $ L^p $-exact controllability of partial differential equations with nonlocal terms. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021053
[2]

Abdelmouhcene Sengouga. Exact boundary observability and controllability of the wave equation in an interval with two moving endpoints. Evolution Equations and Control Theory, 2020, 9 (1) : 1-25. doi: 10.3934/eect.2020014
[3]

Zhen-Zhen Tao, Bing Sun. Galerkin spectral method for elliptic optimal control problem with $L^2$-norm control constraint. Discrete and Continuous Dynamical Systems - B, 2022, 27 (8) : 4121-4141. doi: 10.3934/dcdsb.2021220
[4]

Peter I. Kogut, Olha P. Kupenko. On optimal control problem for an ill-posed strongly nonlinear elliptic equation with $p$-Laplace operator and $L^1$-type of nonlinearity. Discrete and Continuous Dynamical Systems - B, 2019, 24 (3) : 1273-1295. doi: 10.3934/dcdsb.2019016
[5]

Zhen-Zhen Tao, Bing Sun. Error estimates for spectral approximation of flow optimal control problem with $ L^2 $-norm control constraint. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022030
[6]

Guozhen Lu, Yunyan Yang. Sharp constant and extremal function for the improved Moser-Trudinger inequality involving $L^p$ norm in two dimension. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 963-979. doi: 10.3934/dcds.2009.25.963
[7]

Lijuan Wang, Qishu Yan. Optimal control problem for exact synchronization of parabolic system. Mathematical Control and Related Fields, 2019, 9 (3) : 411-424. doi: 10.3934/mcrf.2019019
[8]

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
[9]

Ali Wehbe, Marwa Koumaiha, Layla Toufaily. Boundary observability and exact controllability of strongly coupled wave equations. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1269-1305. doi: 10.3934/dcdss.2021091
[10]

Karina Samvelyan, Frol Zapolsky. Rigidity of the ${{L}^{p}}$-norm of the Poisson bracket on surfaces. Electronic Research Announcements, 2017, 24: 28-37. doi: 10.3934/era.2017.24.004
[11]

Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control and Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017
[12]

Thierry Horsin, Peter I. Kogut. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I. Existence result. Mathematical Control and Related Fields, 2015, 5 (1) : 73-96. doi: 10.3934/mcrf.2015.5.73
[13]

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
[14]

Stanisław Migórski. A note on optimal control problem for a hemivariational inequality modeling fluid flow. Conference Publications, 2013, 2013 (special) : 545-554. doi: 10.3934/proc.2013.2013.545
[15]

Leszek Gasiński. Optimal control problem of Bolza-type for evolution hemivariational inequality. Conference Publications, 2003, 2003 (Special) : 320-326. doi: 10.3934/proc.2003.2003.320
[16]

Giuseppe Da Prato. An integral inequality for the invariant measure of some finite dimensional stochastic differential equation. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3015-3027. doi: 10.3934/dcdsb.2016085
[17]

Jianhui Huang, Xun Li, Jiongmin Yong. A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon. Mathematical Control and Related Fields, 2015, 5 (1) : 97-139. doi: 10.3934/mcrf.2015.5.97
[18]

Piernicola Bettiol. State constrained $L^\infty$ optimal control problems interpreted as differential games. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 3989-4017. doi: 10.3934/dcds.2015.35.3989
[19]

Hee-Dae Kwon, Jeehyun Lee, Sung-Dae Yang. Eigenseries solutions to optimal control problem and controllability problems on hyperbolic PDEs. Discrete and Continuous Dynamical Systems - B, 2010, 13 (2) : 305-325. doi: 10.3934/dcdsb.2010.13.305
[20]

Jan-Hendrik Webert, Philip E. Gill, Sven-Joachim Kimmerle, Matthias Gerdts. A study of structure-exploiting SQP algorithms for an optimal control problem with coupled hyperbolic and ordinary differential equation constraints. Discrete and Continuous Dynamical Systems - S, 2018, 11 (6) : 1259-1282. doi: 10.3934/dcdss.2018071

2021 Impact Factor: 1.141

Tools

Metrics

  • PDF downloads (162)
  • HTML views (65)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy