| Issue |
ESAIM: COCV
Volume 26, 2020
|
|
|---|---|---|
| Article Number | 68 | |
| Number of page(s) | 27 | |
| DOI | https://doi.org/10.1051/cocv/2019052 | |
| Published online | 21 September 2020 | |
On the partial controllability of SDEs and the exact controllability of FBSDES*
1
School of Mathematics and Statistics, Southwest University,
Chongqing
400715, PR China.
2
School of Mathematics, Shandong University,
Jinan
250100, PR China.
** Corresponding author: yuzhiyong@sdu.edu.cn
Received:
8
September
2018
Accepted:
25
August
2019
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.
Mathematics Subject Classification: 93B05 / 93E20 / 60H10
Key words: Stochastic differential equation / partial controllability / rank criterion / observability inequality
This work is supported in part by the National Natural Science Foundation of China (11871310, 11801467, 11701470), the Chongqing Natural Science Foundation (cstc2018jcyjAX0148), the Fundamental Research Funds for the Central Universities (XDJK2017C054), and the Fundamental Research Funds of Shandong University (2017JC016).
© EDP Sciences, SMAI 2020
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