Technical Report on Optimal Linear Multiple Estimation for Landmark-Based Planning via Control Synthesis

The paper’s Proposition 4 solves min_{Wi} K̄2 ΣW K̄2^T with ΣW=∑i Wi Σi Wi^T and ∑i Wi=I, yielding Wi=((∑j Σj^{-1})^{-1}) Σi^{-1}, independent of K̄2 (their eqs. (19)–(20), with a KKT/regularization argument in Appendix B) . The candidate solution proves the same optimizer by a different route: a Loewner-order lower bound ΣW ⪰ (∑i Σi^{-1})^{-1}, then shows trace optimality for any M=K̄2^T K̄2 ⪰ 0 and characterizes uniqueness when M ≻ 0. Both reach the same weights and independence of K̄2; the paper does not discuss uniqueness, whereas the model does.