# American Institute of Mathematical Sciences

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

## Exact controllability of linear stochastic differential equations and related problems

 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.

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:
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##### 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.
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