On the partial controllability of SDEs and the exact controllability of FBSDES

Wang, Yanqing; Yu, Zhiyong

ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 68.

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A notion of partial controllability (also can be called directional controllability or output controllability) is proposed for linear controlled (forward) stochastic differential equations (SDEs), which characterizes the ability of the state to reach some given random hyperplane. It generalizes the classical notion of exact controllability. For time-invariant system, checkable rank conditions ensuring SDEs’ partial controllability are provided. With some special setting, the partial controllability for SDEs is proved to be equivalent to the exact controllability for linear controlled forward-backward stochastic differential equations (FBSDEs). Moreover, we obtain some equivalent conclusions to partial controllability for SDEs or exact controllability for FBSDEs, including the validity of observability inequalities for the adjoint equations, the solvability of some optimal control problems, the solvability of norm optimal control problems, and the non-singularity of a random version of Gramian matrix.

[1] V. Barbu and L. Tubaro, Exact controllability of stochastic differential equations with multiplicative noise. Systems Control Lett. 122 (2018) 19–23. | DOI | MR | Zbl

[2] A.E. Bashirov and N.I. Mahmudov, On concepts of controllability for deterministic and stochastic systems. SIAM J. Control Optim. 37 (1999) 1808–1821. | DOI | MR | Zbl

[3] A.E. Bashirov, H. Etikan and N. Şemi, Partial controllability of stochastic linear systems. Int. J. Control 83 (2010) 2564–2572. | DOI | MR | Zbl

[4] R. Buckdahn, M. Quincampoix and G. Tessitore, A characterization of approximately controllable linear stochastic differential equations, in Stochastic partial differential equations and applications—VII, vol. 245 of Lect. Notes Pure Appl. Math. Chapman & Hall/CRC, Boca Raton, FL (2006) 53–60. | DOI | MR | Zbl

[5] J.-M. Coron, Control and nonlinearity, vol. 136 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (2007). | MR | Zbl

[6] F. Dou and Q. Lü, Partial approximate controllability for linear stochastic control systems. SIAM J. Control Optim. 57 (2019) 1209–1229. | DOI | MR | Zbl

[7] M. Ehrhardt and W. Kliemann, Controllability of linear stochastic systems. Systems Control Lett. 2 (1982/83) 145–153. | DOI | MR | Zbl

[8] N. El Karoui, S. Peng and M.C. Quenez, Backward stochastic differential equations in finance. Math. Finance 7 (1997) 1–71. | DOI | MR | Zbl

[9] B. Gashi, Stochastic minimum-energy control. Syst. Control Lett. 85 (2015) 70–76. | DOI | MR | Zbl

[10] D. Goreac, A Kalman-type condition for stochastic approximate controllability. C. R. Math. Acad. Sci. Paris 346 (2008) 183–188. | DOI | MR | Zbl

[11] E.B. Lee and L. Markus, Foundations of optimal control theory. John Wiley & Sons, Inc., New York, London, Sydney (1967). | MR | Zbl

[12] 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 | MR | Zbl

[13] Q. Lü, J. Yong and X. Zhang, Representation of Itô integrals by Lebesgue/Bochner integrals. J. Eur. Math. Soc. 14 (2012) 1795–1823. | DOI | MR | Zbl

[14] J. Ma and J. Yong, Forward-backward stochastic differential equations and their applications, vol. 1702 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (1999). | MR | Zbl

[15] N.I. Mahmudov, Controllability of linear stochastic systems. IEEE Trans. Automat. Control 46 (2001) 724–731. | DOI | MR | Zbl

[16] N.I. Mahmudov, Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces. SIAM J. Control Optim. 42 (2003) 1604–1622. | DOI | MR | Zbl

[17] D. Nualart, The Malliavin calculus and related topics, Probability and its Applications. Springer-Verlag, Berlin (2006). | MR | Zbl

[18] S. Peng, Backward stochastic differential equation and exact controllability of stochastic control systems. Progr. Nat. Sci. 4 (1994) 274–284. | MR

[19] M. Sîrbu and G. Tessitore, Null controllability of an infinite dimensional SDE with state- and control-dependent noise. Syst. Control Lett. 44 (2001) 385–394. | DOI | MR | Zbl

[20] Y. Sunahara, T. Kabeuchi, Y. Asada, S. Aihara and K. Kishino, On stochastic controllability for nonlinear systems. IEEE Trans. Automatic Control AC-19 (1974) 49–54. | DOI | MR | Zbl

[21] 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 | MR | Zbl

[22] 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 | MR | Zbl

[23] Y. Wang and C. Zhang, The norm optimal control problem for stochastic linear control systems. ESAIM: COCV 21 (2015) 399–413. | Numdam | MR | Zbl

[24] 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 | MR | Zbl

[25] Z. Wu and Z. Yu, Backward stochastic viability and related properties on Z for BSDEs with applications. J. Syst. Sci. Complex. 25 (2012) 675–690. | DOI | MR | Zbl

[26] J. Zabczyk, Controllability of stochastic linear systems. Syst. Control Lett. 1 (1981/82) 25–31. | DOI | MR | Zbl