Role of stochastic controllability quantities in other stochastic control problems

Investigate whether the stochastic controllability matrix and stochastic Gramian introduced for the open-loop stochastic minimum-energy control problem play a role in other types of stochastic control problems.

Background

The paper introduces a stochastic controllability matrix and stochastic Gramian as counterparts of the deterministic controllability matrix and Gramian for open-loop minimum-energy control of stochastic linear time-invariant systems. The authors establish that these quantities characterize the structure and optimal cost of the stochastic minimum-energy control solution, but do not determine whether the same quantities are useful in broader classes of stochastic control problems. The conjectured broader applicability is therefore left unresolved.