Global existence and uniqueness of the LEQG covariance-steering solution branch

Establish global existence and uniqueness of the LEQG covariance-steering solution branch by upgrading the partial-Jacobian analysis to the nonzero-risk-sensitivity expression and proving that the implicit-function mapping h can be extended in class C^1 to the boundary of every finite interval.

Background

The paper formulates the linear exponential quadratic Gaussian (LEQG) covariance-steering problem and derives a coupled Riccati boundary-value formulation for its optimal linear feedback controller. Under the matched noise and input channel condition, Theorem 1 establishes only a local C1 solution branch in the risk-sensitivity parameter theta around the risk-neutral solution theta = 0.

The authors identify a possible route to a global result: use the more general partial Jacobian formula valid for nonzero theta, which includes an integral correction term, and then show that the implicit-function solution mapping h remains extendable in C1 up to the boundary of any finite theta interval. Whether this program succeeds is left unresolved.

References

While Theorem \ref{thm:LocalExistenceUniqueness} proved local existence-uniqueness of solution, we suspect this result can be made global by upgrading PartialJacobian with PartialJacobianAttthetaNotEqualToZero, and then showing that $h$ can be extended in $\mathcal{C}{1}$ to the boundary of any finite interval. This remains ongoing work.

— Linear Exponential Quadratic Gaussian Covariance Steering  (2609.12463 - Cherian et al., 11 Sep 2026) in Section 5, Concluding Remarks